Aero and Vibroacoustics of Automotive Turbochargers [1 ed.] 978-3-642-35069-6, 978-3-642-35070-2

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Aero and Vibroacoustics of Automotive Turbochargers

Hung Nguyen-Schäfer

Aero and Vibroacoustics of Automotive Turbochargers

123

Hung Nguyen-Schäfer Bosch Mahle Turbo Systems GmbH Stuttgart Germany

ISBN 978-3-642-35069-6 DOI 10.1007/978-3-642-35070-2

ISBN 978-3-642-35070-2

(eBook)

Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012955042 Ó Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science?Business Media (www.springer.com)

Silence is a source of great strength. (In der Ruhe liegt die starke Kraft.) Lao Tzu (6th century BCE)

Preface

This book has arisen from my many years of experience in the automotive industry. It is intended for advanced undergraduates in mechanical engineering, research scientists, and practicing engineers working on automotive turbochargers. It could be also used as an aero and vibroacoustics textbook in colleges and universities, and as a practical handbook of Aero and Rotordynamics in turbocharger industries. Aero and Vibroacoustics of Automotive Turbochargers is a widely interdisciplinary working field dealing with noise source generation and airborne noise propagation from turbochargers to the environment. First, it involves thermodynamics of turbomachinery to design the turbocharger and to determine its working conditions. Second, it involves aerodynamics and the study of flow dynamics and instabilities of rotating stall and surge in compressors, which lead to aerodynamic noise. These are self-excited vibration noises rather than the response of external excitation noise sources. Third, it involves fluid and rotordynamics to study the unbalance whistle due to rotor unbalance, and self-excitation flow instabilities of oil whirls in rotating floating ring bearings, leading to the constant tone (howling). Both noise types are called rotordynamic noise. Lastly, it involves the noise propagation computation based on the Lighthill’s analogy to investigate the airborne noise of turbochargers in passenger vehicles. However, some mathematical backgrounds in vector, matrix, and tensor analysis are required to comprehend this interdisciplinary working field of Aero and Vibroacoustics. Airborne noise has recently become a much more important topic in passenger cars next to CO2 reduction, and reducing fuel consumption. In recent years, engine noise has become relatively quiet so that other noise sources now become audible in passenger vehicles. Drivers and passengers want to use online communications via Internet and mobile phones, or relax with audio and video infotainment media without any external noise disturbance during driving. Therefore, airborne noise level in the cabin is required to be as low as possible. Moreover, car frames made of light metals and light periphery components have been used to reduce the vehicle mass, and to save fuel consumption and reduce emissions. Under these circumstances, induced noise of turbochargers excites the periphery components vii

viii

Preface

neighboring the turbocharger, leading to additional airborne noises in the car cabin and environment. The root causes of different airborne noise types in turbochargers, such as pulsation, rotational, growling, whining noise, unbalance whistle, constant tone, crackling noise, high-order harmonic noise, and wear noise are thoroughly studied. Both active and passive measures are recommended to reduce the airborne noise levels in passenger vehicles. Despite all my careful efforts, there may be some unpredictable errors in this book. I would be very grateful to receive your feedback and hints of errors. Therefore, readers of this book need to have a thorough analysis before applying it to their individual applications, and take their own responsibilities for possible damages. I am grateful to Dr. Martin Knopf, the executive director of Bosch Mahle Turbo Systems (BMTS) in Stuttgart, for all his support and encouragement. Especially, I am also indebted to my colleague Daren Bolbolan for reviewing this book, and giving me constructive and helpful suggestions. In addition, I would like to acknowledge Dr. Jan-Philip Schmidt at Springer in Heidelberg for proofing mathematical derivations and for editing this book; Mrs. Heather King also at Springer and Dr. Shine David at SPS (P) Ltd. for invaluable recommendations and helps in publishing this book. Finally, my special thanks go to my wife for her understanding, patience, and endless support during writing this book in my leisure time and on vacation. Stuttgart, Germany

Hung Nguyen-Schäfer

Contents

1

Introduction to Turbocharging . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Applying Turbochargers to Downsized Engines 1.2.1 Single-Stage Turbochargers with EGR. . 1.2.2 Biturbochargers. . . . . . . . . . . . . . . . . . 1.2.3 Two-Stage Turbochargers . . . . . . . . . . 1.2.4 Turbo Compounds. . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 4 4 6 7 9 9

2

Induced Noise Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 14

3

Acoustic Propagation Theory . . . . . . . . . . . . . . . . . . . . . 3.1 Aeroacoustic Characteristics . . . . . . . . . . . . . . . . . . . 3.2 Acoustic Propagation Equations . . . . . . . . . . . . . . . . 3.3 Computation of Acoustic Propagations. . . . . . . . . . . . 3.3.1 Computational Methods . . . . . . . . . . . . . . . . . 3.3.2 BEM for Aeroacoustics . . . . . . . . . . . . . . . . . 3.3.3 FEM for Aeroacoustics . . . . . . . . . . . . . . . . . 3.4 Frequency Modulations of Nonlinear Aeroacoustics. . . 3.4.1 Responses of Nonlinear Aeroacoustic Systems . 3.4.2 Modulated Noise Sideband Frequencies . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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15 15 22 27 27 29 32 35 36 37 44

4

Analyzing Root Causes of Noise . . . . . . . . . . . . . 4.1 Aerodynamic Noise . . . . . . . . . . . . . . . . . . . 4.1.1 Root Causes of Aerodynamic Noise . . 4.1.2 Aerodynamics of Radial Compressors . 4.1.3 Stall and Surge in Radial Compressors

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ix

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Contents

4.2

Unbalance Whistle. . . . . . . . . . . . . . . . . . 4.2.1 Root Causes of Unbalance Whistle . 4.2.2 Unbalance Types of Turbochargers . 4.2.3 Unbalance Change . . . . . . . . . . . . 4.2.4 ICM Trim Balancing . . . . . . . . . . . 4.3 Subsynchronous Constant Tone . . . . . . . . . 4.4 High-Order Harmonic Noise . . . . . . . . . . . 4.5 Wear Noise. . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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63 63 64 66 66 68 69 70 72

5

Computational Nonlinear Rotordynamics of Turbochargers 5.1 Vibration Equations of Automotive Turbocharger Rotors . 5.2 Two-Phase Reynolds Lubrication Equation . . . . . . . . . . . 5.3 Nonlinear Bearing Forces . . . . . . . . . . . . . . . . . . . . . . . 5.4 Boundary Conditions of Nonlinear Rotordynamics . . . . . 5.5 Computational Results of Turbochargers . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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73 73 75 77 79 80 86

6

Subsynchronous Constant Tone . . . . . . . . . . . . 6.1 Transmitting Way of Airborne Noise . . . . . . 6.2 Constant Tone Root Causes . . . . . . . . . . . . 6.3 Constant Tone Analysis . . . . . . . . . . . . . . . 6.4 Calculating Ring Speed Ratio . . . . . . . . . . . 6.5 Pre and Post Constant Tone . . . . . . . . . . . . 6.6 Measures of Reducing Constant Tone Level . 6.6.1 Active Measures . . . . . . . . . . . . . . . 6.6.2 Passive Measures . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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89 90 91 95 97 102 103 103 105 108

7

Eigenfrequency Modifications to Reduce Constant Tone Level. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 112

Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115

Appendix A: Thermodynamic Characteristics of Perfect Gases . . . . . .

117

Appendix B: Transformation of Coordinates . . . . . . . . . . . . . . . . . . .

119

Appendix C: Solutions of the Characteristic Equation with Complex Coefficients . . . . . . . . . . . . . . . . . . . . . . .

123

Appendix D: Stability Condition of Linear Aeroacoustics . . . . . . . . . .

125

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Contents

xi

Appendix E: Routh–Hurwitz Stability Criterion . . . . . . . . . . . . . . . . .

129

About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133

Abbreviations

Notations A a av b CSG c cc cm C0 CW c1 c2 Di, Do e FB FJ F1, F2 Fr , Ft f h I Ip It j KS K k kc

Rotor mass, and inertia moment matrix Acceleration Bunsen’s coefficient External force vector Damping, and gyroscopic matrix Diagonal damping coefficient; radial bearing clearance; absolute velocity; sound speed; damping coefficient Cross-coupled damping coefficient Meridional velocity component Compressor characteristic slope Compressor wheel Radial inner bearing clearance Radial outer bearing clearance Inner and outer bearing diameter Journal eccentricity Bearing force acting on journal Journal force acting on bearing Force components in X1 and X2 Force components in r and t External force vector Oil film thickness Noise intensity Polar mass inertia moment Transversal mass inertia moment Imaginary unit Stiffness coefficient matrix Stiffness coefficient Wavenumber Cross-coupled stiffness coefficient xiii

xiv

L Li, Lo Lp M Ma N NVH p0 p p0 P R RSR Q r0 r s T0 T T0 t u U Ur Ub v w x

Abbreviations

Noise pressure level Inner and outer bearing width Noise power level Rotor mass matrix Mach number Degrees of freedom (DOF) Noise vibration hardness Ambient absolute pressure Oil film pressure Noise pressure Noise power Journal radius Ring speed ratio Noise source Initial gas bubble volumetric fraction Gas bubble volumetric fraction Rotor eigenvalue Ambient temperature Oil film temperature Throttle characteristic slope; perturbed fluid temperature Time Circumferential velocity Unbalance force and moment matrix Rotor circumferential velocity Bearing ring circumferential velocity Gas velocity Relative velocity Vibration response

Greek Symbols a b c e f u k g gTC gliq gmix gi go

Attitude angle Blade angle Journal angular position Journal relative eccentricity Damping ratio Flow coefficient Average fluid velocity ratio; eigenvalue Speed ratio Overall turbocharger efficiency Oil viscosity Oil mixture viscosity Dynamic viscosity of inner oil film Dynamic viscosity of outer oil film

Abbreviations

h j q0 w x X XR

Bearing filling grade Isentropic exponent of gas Perturbed fluid density Pressure-rise coefficient Whirl frequency Rotor angular speed Bearing-ring angular speed

xv

Chapter 1

Introduction to Turbocharging

1.1 Introduction Downsized engines, reducing the number of cylinders or volumetric size of cylinders, combined with the use of exhaust turbochargers, are being applied more and more in the automotive industry in order to comply with the recently enacted emission laws for CO2 and nitrogen oxides (NOx) reduction in automotive vehicles. These downsized engines with fewer numbers of cylinders or small cylinder volume induce less friction power between the pistons and cylinders. Additionally, total weight of the vehicle is also reduced due to a smaller engine, leading to less driving friction forces. Both measures help to reduce emissions and improve the specific power of small engines [1]. The fuel consumption of passenger vehicles could be reduced by nearly 18 % without loss of engine power through the combined use of a turbocharger (TC), high-pressure direct injection (HPDI), low-pressure exhaust gas recirculation (LP-EGR), variable compression ratio (VCR), and variable valve train (VVT). With extreme engine downsizing, using turbochargers and hybrid techniques, the reduction of fuel consumption could reach 30 % in total. Such measures could help the automotive industry reduce the enacted average emission law for CO2, reduced from 120 g/km in 2012 to 95 g/km in 2020 for new passenger vehicles; and from 175 g/km in 2014 to 147 g/km in 2020 for new light duty commercial vehicles in the European Union (EU) countries, as shown in Figs. 1.1 and 1.2. After the combustion of fuel in the engine, the large energy of exhaust gas still remains in the form of enthalpy from the engine outlet because the temperature of exhaust gas is quite high (e.g., diesel 820 – 850 C and gasoline 950 – 1050 C). Generally, this enthalpy energy escapes from the engine to the environment. Why do we not capture it to boost the engine for improving its specific power? To do that, an exhaust-gas turbocharger is necessary to collect the exhaust gas in the turbine, and compress the intake air in the compressor to a high pressure for the engine combustion. This procedure is called turbocharging, which improves

H. Nguyen-Schäfer, Aero and Vibroacoustics of Automotive Turbochargers, DOI: 10.1007/978-3-642-35070-2_1,  Springer-Verlag Berlin Heidelberg 2013

1

2

1 Introduction to Turbocharging

Fig. 1.1 CO2 average emission limits for new passenger vehicles

Passenger Vehicles (PV)

120 g/km CO2

100% new vehicles

65% new vehicles

2012

Fig. 1.2 CO2 average emission limits for new commercial vehicles (LD)

75% new vehicles

2013

80% new vehicles

2014 Year

95 g/km CO2 100% new vehicles

2015

2020

Light Duty Commercial Vehicles (LDCV) 175 g/km CO2

70% new vehicles

2014

80% new vehicles

2015

90% new vehicles

2016

100% new vehicles

2017

147 g/km CO2 100% new vehicles

2020

Year

the engine-specific power. The exhaust-gas turbocharger consists of the CHRA (center housing and rotating assembly), as shown in Fig. 1.3, turbine, compressor, and actuator [2]. Both compressor and turbine wheels are mounted and fixed in the rotor shaft that is supported on the bearing system including two separate rotating floating radial bearings, and the thrust bearing. The exhaust gas expands in the turbine wheel and generates the rotational kinetic energy for the rotor shaft. The created turbine energy drives the rotor shaft and compresses the charge air to a high pressure by means of the compressor. The concept of using exhaust-gas turbochargers takes advantages of improving specific engine power and as well as reducing CO2 and NOx. Enabling either having the same power of the original engine but less fuel consumption, which reduces carbon dioxide and as well as nitrogen oxides in the downsized engine or achieving more power at the same fuel consumption in the turbocharged engine without downsizing for those who need more driving fun. However, the first one is an important key to help us comply with the new emission law for the automotive vehicles (see Sect. 1.2). In this chapter, automotive turbochargers are applied to downsized engines to comply with the given reduction of CO2 emission from 2012. Due to car

1.1 Introduction compressor wheel

3 thrust bearing

radial bearings

turbine wheel

Fig. 1.3 Automotive turbocharger’s CHRA (Courtesy BMTS)

infotainment and online communications via Internet while driving, induced airborne noise in the passenger cabin is required to be kept as low as possible, especially in passenger vehicles. Additionally, light car frames and peripheral components are used to reduce total vehicle mass; in turn, to save fuel consumption (mpg), leading to reduction of exhaust gas emissions. Under such boundary conditions, noise induced by the engine and turbocharger is transmitted through the turbocharger, air filter, air intercooler, exhaust gas system (catalyzer, diesel particle filter, and muffler), and car frame into the passenger cabin and car environment. This transmitted noise is called airborne noise that should be strongly reduced in passenger vehicles. Besides engine noise, some synchronous and asynchronous (super- and subsynchronous) noise types induced in automotive turbochargers are caused by the flow instabilities (rotating stall and surge) in the compressor wheel, rotation of compressor and turbine wheels, rotor unbalance, oil whirls in hydrodynamic radial bearings, and outer race clearance in rolling element bearings. The airflow noise is defined as the aerodynamic noise (aeroacoustics). The unbalance whistle, constant tone in the oil-film radial bearings, and high-order harmonic noise in the rolling element bearings are called the rotordynamic noise (vibroacoustics or rotordynamic acoustics). Due to nonlinearity of the automotive turbocharger’s bearing characteristics, non-synchronous rotor responses are induced from superimposing the synchronous rotor unbalance excitation (1X) on subsynchronous components of the nonlinear rotor system, such as inner and outer oil whirl vibrations [2, 3]. In turn, the inner oil whirl generates subsynchronous noise that is usually called ‘‘constant tone’’, a self-excited vibration response. Necessary measures must be applied to turbochargers in order to minimize these turbocharger noises in passenger vehicles. For this reason, aerodynamics in radial compressors and nonlinear rotordynamic responses of the rotor have been thoroughly studied in a passenger vehicle’s turbocharger running on two separate rotating floating ring bearings. Computational results and measurements give a hint that inner oil whirl is the root cause of constant tone. The induced noise of the turbocharger excites other neighboring components nearby and is transmitted to the vehicle cabin and environment.

4

1 Introduction to Turbocharging

1.2 Applying Turbochargers to Downsized Engines In automotive applications, single-stage turbochargers are normally used at the air compression ratio to nearly 2.5. There are two layouts of turbochargers with exhaust gas recirculation (EGR) at low pressure (LP-EGR) and high pressure (HPEGR). By means of engine exhaust gas recirculation, the burning temperature spike of exhaust gas is lowered, leading to reducing NOx. Generally, turbocharged downsized engines with the same original power could save nearly 10 % fuel consumption by reducing the cylinder volume by 25 %. To abide by the current emission law or to boost engines of passenger vehicles, two-stage turbochargers are applied to engines working at large charge-air compression ratios above 4. Two-stage turbochargers consist of two single-stage turbochargers with different sizes that are sequentially set up in the engine. Some applications of different turbocharger types are discussed in the following subsections.

1.2.1 Single-Stage Turbochargers with EGR Figure 1.4 shows a single-stage turbocharger with HP-EGR in which the exhaust gas partly returns to the engine with a maximum of 50 % EGR rate before entering the turbine. The EGR rate is defined by the ratio of the recirculation mass flow rate to the outflow mass flow rate of the exhaust gas. The EGR valve regulates the EGR mass flow rate so that NOx emission fulfills the current emission law. Due to the expansion energy of the exhaust gas, the turbine (T) propels the compressor (C) that compresses the intake air to the maximum pressure ratio of nearly 2.5. The compression process in the compressor brings the inlet air from the ambient condition to high pressure and temperature as well. To maintain the compressed Fig. 1.4 A single-stage turbocharger with HP-EGR Engine

EGR intercooler

intercooler

C

T

air

exh. gas

1.2 Applying Turbochargers to Downsized Engines

5

Fig. 1.5 Single-stage turbochargers with LP-EGR

Engine

intercooler

T

C intercooler

air

EGR

soot trap

exh. gas

charge air with large mass flow rate for the engine combustion, the intercooler cools down the charge air temperature after the compressor. Otherwise, the air mass flow rate would be reduced due to low charge air density at higher temperatures. In case of turbochargers with HP-EGR, the recirculated exhaust gas pressure after the intercooler must be higher than the charge air pressure at the engine inlet. Hence, the pressure ratio of the turbine is required high enough to overcome the charge air pressure. Generally, the turbine with HP-EGR is designed to be smaller than the turbine without HP-EGR so that the exhaust gas pressure remains higher than the compressed charge air pressure. Therefore, the engine must work against the high exhaust gas pressure at the engine outlet. As a result, the fuel consumption increases compared to the turbocharging system without HP-EGR, especially at high turbocharging pressures. That is the reason why the HP-EGR valve should be closed as soon as the exhaust gas meets the requirements of the current emission law. To overcome this disadvantage, the HP-EGR valve is replaced in the low-pressure site at the turbine outlet. The turbocharger layout shown in Fig. 1.5 is called turbocharger with LP-EGR. Its advantage is that the engine working condition against the high exhaust gas pressure drops; therefore, the fuel performance becomes much more efficient. However, the LP-EGR turbocharger has two disadvantages: first, the low exhaust gas pressure of the turbine outlet and pressure drop in the soot trap limit the LP-EGR mass flow rate, hence the EGR rate; second, the exhaust gas and ambient air mix together at the compressor inlet; they are compressed in the compressor to higher pressure. Due to the pressure increase in the compressor wheel, the exhaust gas condenses, leading to erosion on the surface of the wheel. Additionally, the unburned hard particles in the exhaust gas impact and damage the aluminum compressor wheel at high rotor speeds, especially the blades at the inlet of the compressor wheel. To prevent damage to the compressor wheel, nickel phosphor (NiP ) coating layer with HV from 600 to 1,000 is usually applied to the compressor wheel. However, this causes a reduction in the mass flow rate of the charge air, and therefore the compressor power as well. Due to low-cycle fatigue (LCF) of the driving cycle, the compressor wheel lifetime is reduced.

6

1 Introduction to Turbocharging

Fig. 1.6 Bi-/TwinTurbochargers with using intercooler

Engine

blow-off valve intercooler

C

air

shut-off valves

T

T

exh. gas

C

air

1.2.2 Biturbochargers Bi-/Twin-turbochargers consist of two small turbochargers of the same volumetric size; they are parallel setups and operate at the same time in the entire rotor speed range (parallel bi-/twin-turbo). The mass flow rate of exhaust gas is divided into both turbines of the bi-turbo at opening both shut-off valves. In case of an engine with four cylinders, two cylinders provide exhaust gas for each turbocharger, as shown in Fig. 1.6. After compression and cooling, both compressors provide the engine with the total compressed charge air. With small geometry of turbochargers, the rotor generally has small mass inertia moment that speeds up quickly to achieve maximum torque; therefore, the transient behavior of the parallel bi-turbo is strongly improved in the low-end torque (LET). However, a small turbocharger delivers too little air mass flow rate alone and does not fulfill the required nominal engine power. Hence, both small turbochargers have to operate in parallel in order to deliver enough compressed charge air for the required nominal engine power. In case of the sequential bi-turbo that is a parallel setup and sequentially operating, only one of the bi-turbo, e.g., the left one operates alone at the low engine speed by closing both shut-off valves. The small turbocharger builds up the charge air pressure much earlier and improves the transient behavior of the turbocharger at low engine speeds. At high engine speeds, the other turbocharger (i.e. the right one) is additionally turned on by opening both shut-off valves where both turbochargers work parallel like the parallel bi-turbo. Hence, the mass flow rate of the charge air increases to empower the engine at high rotor speeds. In strongly powered W-engines with 16 cylinders (e.g. Bugatti Veyron 16.4 Super Sport with a power of nearly 1,200 hp), four single turbochargers (sequential quad-turbo) are used; every turbocharger is applied to four cylinders (4 9 4).

1.2 Applying Turbochargers to Downsized Engines Fig. 1.7 Regulated twostage turbochargers with HPEGR

7

Engine

EGR blow-off valve intercoolers

C-bypass

C1

T1 T-bypass

primary TC

intercooler

secondary TC

C2

air

T2

exh. gas

1.2.3 Two-Stage Turbochargers Two-stage turbochargers are used for high-pressure ratios in which two different volumetric sizes of turbochargers are sequentially set up and work at the given procedure regulated by the bypass valves (see Fig. 1.7). Leaving the engine, the exhaust gas containing large enthalpy is at high pressure and temperature; therefore, the turbocharger at the primary stage (C1, T1) is smaller than the secondary stage (C2, T2). In case of unregulated pressures in sequentially working condition for both stages in commercial vehicle applications, the primary turbocharger is normally about 15 % smaller than the single-stage turbocharger; the secondary turbocharger is 15 % larger than the single-stage one. At low engine speeds, only the small turbocharger works alone because of its small mass inertia moment; therefore, the transient behavior is significantly improved compared to the large one. During this time, the bypass valves of the primary turbocharger are closed, and the waste gate of the secondary turbine is fully opened so that the secondary turbocharger is nearly decoupled from the two-stage turbocharger. At middle engine speeds from about 1,500 rpm, both turbochargers work sequentially by means of the regulated bypass valves. As soon as the required charge air pressure is reached at high engine speeds, only the secondary turbocharger works alone because the larger turbine (T2) has large efficiency at high rotor speeds. In this case, the primary turbocharger will be decoupled from the two-stage turbocharger, in which both bypass valves of the primary turbocharger are fully opened.

8 Fig. 1.8 Compression pressure ratios of regulated two-stage turbochargers

1 Introduction to Turbocharging

πC Full load curve

π C , 2– stage π C ,secondary

π C, primary

1 •

mred

Fig. 1.9 Engine torques of single- and two-stage turbochargers

M two-stage TC secondary stage TC

single-stage TC

primary stage TC

NTC

Similar to the earlier turbochargers, the compressed charge air temperature must be cooled down by air intercoolers before entering the engine in order to maintain the high mass flow rate of the intake air for the required nominal engine power. The compressor pressure ratio in the full load curve of the two-stage turbocharger is displayed in Fig. 1.8. This pressure ratio results from the product of the pressure ratios of two compressors of the two-stage turbocharger. Due to the small volumetric size of the primary turbocharger, its pressure load curve speeds up faster at low engine speeds to achieve a good transient response compared to the transient response of the secondary turbocharger in LET. The engine torques of the single- and two-stage turbochargers versus the rotor speed of turbocharger are compared to each other, as illustrated in Fig. 1.9. The single-stage turbocharger is larger than the primary but smaller than the secondary stage of the two-stage turbocharger. Therefore, the transient response of the twostage turbocharger is better than the single-stage turbocharger. The advantages of the two-stage turbocharger are the better response behavior, higher compression ratio of the charge air, and higher engine torque in the operating speed range.

1.2 Applying Turbochargers to Downsized Engines

9

Fig. 1.10 Turbo compounds with a power turbine (PT) Engine gear box

PT EGR intercooler

intercooler

C

T

air

exh. gas

1.2.4 Turbo Compounds Turbo compounds consist of a turbocharger and power turbine (PT) that uses the exhaust gas from the turbocharger, still at high temperature, to generate additional engine power for the engine. The power turbine shaft is directly geared to the engine drive shaft to increase the engine torque and power to help to increase the engine efficiency, as shown in Fig. 1.10. Turbo compounds are possibly applied in commercial vehicles to gain some additional driven energy to the engine shaft.

References 1. Golloch, R.: Downsizing bei Verbrennungsmotoren (in German), Springer, Berlin (2005) 2. Nguyen-Schäfer, H.: Rotordynamics of Automotive Turbochargers. Springer, Berlin (2012) 3. Nguyen-Schäfer, H., Kleinschmidt, R.: Analysis and nonlinear rotordynamics computation of constant tone in automotive turbochargers. 17th ATK Conference, Dresden (2012)

Chapter 2

Induced Noise Types

Noises, as shown in Fig. 2.1 induced in automotive turbochargers are normally classified into the following different types of noises [1, 2]. Pulsation whistle is caused by slight differences in the compressor wheel chamber volumes due to milling or molding process variations. Its frequency ranges from 1,200 to 4,500 Hz and occurs at engine speeds from 1,500 to 3,500 rpm, usually in second gear with high load. It is also dependent of the number of these unequal compressor wheel chambers. Rotational noise (rotating-blade-related noise) is generated by the rotating blades of the compressor or turbine wheels at an engine speed range from 1,400 to 2,500 rpm in second and third gears. This noise has very high frequencies between about 8 and 18 kHz or higher, which result from the number of blades and rotor speed. Although adult human ears cannot recognize noise higher than 16 kHz, animals in the car could hear such high frequencies. Growling noise (compressor-stall-related noise) is induced by the partial reversal of the charge air in the compressor wheel. Partial flow separation at the suction side near the blade outlet causes the growling noise. Its frequency ranges in a wide band between 1,200 and 3,500 Hz containing partly metallic noise, and occurs at engine speeds between 1,400 and 2,500 rpm in second and third gears. Growling noise propagates in the direction of the compressor outlet, and the charge-air intercoolers. Whining noise (compressor-surge-related noise) is induced by a deep surge condition in the compressor wheel where the charge air totally recirculates from the compressor outlet to compressor inlet. The whining noise occurs when driver suddenly releases the gas pedal, causing the required charge-air mass flow rate for the engine to be suddenly reduced at still high turbocharger speeds. This leads to the deep surge working condition in the compressor. Its frequency ranges in a wide band from nearly 800 to 2,700 Hz containing partly metallic noise, and occurs at engine speeds from 1,400 to 2,500 rpm in second and third gears. Whining noise propagates in the direction of the compressor inlet and the air filter system. Unbalance whistle is caused by the unbalanced rotor and unbalance change of the rotor after a long-term operating time. Its harmonic frequency (1X) is between H. Nguyen-Schäfer, Aero and Vibroacoustics of Automotive Turbochargers, DOI: 10.1007/978-3-642-35070-2_2, Ó Springer-Verlag Berlin Heidelberg 2013

11

12

2 Induced Noise Types

Pulsation noise

Unbalance whistle

Crackling noise

Rotational noise

Growling noise

Whining noise

Constant tone (oil-film bearings)

High-order harmonic noise (ball bearings)

Wear noise (ball bearings)

Fig. 2.1 Induced noises of automotive turbochargers

1,200 and 4,500 Hz happening at an engine speed range between 1,500 and 3,500 rpm in second gear at relatively low loads. Constant tone (howling) is induced by the inner oil whirl due to oil whirling in the oil-film radial bearing; its frequency lies between 600 and 1,000 Hz in the human audible range. The inner oil whirl frequency order in the rotating floating ring bearings reduces from about 0.4 to 0.3X as the rotor speed increases. Therefore, the inner oil-whirl induced frequency varies in a small range between 600 and 1,000 Hz, and is generally considered as quasi-constant in the automotive turbocharger operating speed range. Constant tone often occurs at engine speeds between 1,500 and 3,500 rpm in second to fifth gears with middle to high loads. High-order harmonic noise often occurs in turbochargers using rolling element bearings (ball and roller bearings) where the constant tone fortunately does not happen. This noise type is caused by an inappropriate design of the oil damper clearance between the bearing outer race or cartridge and bearing housing [1]. The high-order harmonic noise has multiple harmonic frequency orders of 2, 3, 4, 5X, and above, as well as modulation sideband frequencies. Wear noise mostly occurs in turbochargers using rolling element bearings (ball and roller bearings) if wear defects of balls, rollers, cage, inner, and outer races take place. The wear noise has different asynchronous frequencies that will be discussed in Sect. 4.5. Crackling noise mostly takes place in wastegated turbochargers in which the waste gate (WG) vibrates about its pivot axis. This vibration generates the metallic crackling noise in various asynchronous frequencies between about 7 and 15 kHz in the WG open position. However, the crackling noise strongly reduces or

2 Induced Noise Types

13

diminishes when the WG is in the closed position due to the damped vibration amplitude at the waste gate seat located in the turbine housing. The first four noise types are caused by aerodynamics in the compressor and turbine wheels. The unbalance whistle is induced by the rotor unbalance at high rotor speeds, and the constant tone is generated in the inner bearing hydrodynamic oil film. Both unbalance whistle and constant tone are therefore called rotordynamic noises. Rotordynamics engineers have studied both unbalanced whistle and constant tone and found that an appropriate two-plane rotor trim balancing at high balancing speeds reduces the unbalance whistle but not the constant tone; however, the constant tone could be released at an extremely large rotor unbalance. In this book, the root causes of these noise types are studied; measures of reducing them are thoroughly discussed. Furthermore, efforts are also made to physically understand the noise behaviors and to reduce the undesirable noise generated in automotive turbochargers. Aerodynamic flows in compressor wheels are used to figure out the causes of the aerodynamic noises of the pulsation whistle, rotational, growling, and whining noises. Rotordynamics is applied to analyze the rotordynamic noises of the unbalance whistle, constant tone, crackling noise, highorder harmonic noise, and wear noise. And finally, some measures are discussed to improve the noise behavior (NVH) of automotive turbochargers in passenger vehicles. Figure 2.2 shows the noise sources and response noises in a typical automotive turbocharger. The aerodynamic noise sources ‘‘1’’ of the pulsation, rotational, growling, and whining noises are transmitted from the turbocharger through the compressor housing to the air filter, air supply pipes, and air intercooler that are excited by the aerodynamic noise, leading to the response noises. The whining noise generally excites the intake-air supply components due to the flow reversal from the compressor outlet to the compressor inlet. On the contrary, the growling noise affects more the charge-air components, such as the charge-air duct and intercooler, as displayed in Fig. 2.2. The partial flow separation near the

: noise source

engine (4 cyl.)

: response noise intercooler

turbine

compressor

C 1

air

T 2

air filter

Fig. 2.2 Airborne noises of automotive turbochargers

exhaust gas system

exh. gas

14

2 Induced Noise Types

compressor outlet is to blame for the growling noise propagating direction in the charge-air components; this will be discussed in detail in Sect. 4.1. The rotordynamic noise sources ‘‘2’’ of the unbalance whistle, crackling noise, and constant tone in oil film bearings/high-order harmonic or wear noise in ball bearings are propagated through the bearing oil films/bearing balls and oil damper, and bearing center housing to the exhaust-gas system of the catalyzer, diesel particle filter (DPF), and muffler. The noise sources also excite the components neighboring the turbocharger, leading to the additional response noises. All induced noises are eventually transmitted through the car frame as airborne noises in the car cabin and environment. The audible airborne noise level depends on the vehicle types and operating engine speeds. Therefore, the delivery and limit acceleration unbalance levels must be determined for each car type; this will be discussed later in Sect. 4.2.

References 1. Nguyen-Schäfer, H.: Rotordynamics of Automotive Turbochargers. Springer, Berlin (2012) 2. Nguyen-Schäfer H., Kleinschmidt, R.: Analysis and nonlinear rotordynamics computation of constant tone in automotive turbochargers. 17th ATK Conference, Dresden (2012)

Chapter 3

Acoustic Propagation Theory

3.1 Aeroacoustic Characteristics Aerodynamic noise types such as pulsation, rotational, growling, and whining noise are generated from airflows in the compressor of turbocharger. Rotordynamic noises are the noise types like unbalance whistle, constant tone, crackling noise, and highorder harmonic noise. The unbalance whistle is caused by rotor unbalance; the constant tone is induced by the inner oil whirl occurring in the oil-film radial bearings due to self-excitation instability. Additionally, the high-order harmonic noise and possible wear noise mostly occur in the rolling element bearings. Induced noise is transmitted through the bearing oil films, bearing center housing, compressor housing, air filter system, charge-air intercoolers, exhaust-gas manifold, exhaust-gas system (catalytic converter, particle diesel filter (DPF), and muffler), and car frame to the cabin, as shown in Fig. 3.1. The induced noise excites the bearing center housing and periphery components near the turbocharger, such as air filter, charge-air intercoolers, and exhaust-gas system of catalyzer, DPF, and muffler. The excited vibration responses emit airborne noise into the turbocharger’s environment. Such airborne noise is uncomfortable and undesirable for the vehicle occupants, and should be reduced as much as possible in passenger-type vehicles. Figure 3.2 shows some wave characteristics of acoustic propagation, such as vibration amplitude n(t), oscillating air-particle velocity v(t), perturbed noise pressure p0 (t), and propagating wave front with a sound speed c. At first, an airparticle oscillates with an angular frequency x and impacts on the neighboring air particles in a compressible fluid (e.g. air). The impact process continues onto the next air particles, leading to propagating waves in the air. This process is compared to the domino effect where the first domino falls with the velocity v; it affects the second one that further affects the next one, and so on. During falling of the domino pieces, the audible noise propagates in the air at the speed of sound c, which is caused by the noise pressure perturbation. In this case, air particles are accelerated and decelerated. This causes a compression wave with v [ 0 at increasing air pressure and density; an expansion wave with v \ 0 at decreasing air

H. Nguyen-Schäfer, Aero and Vibroacoustics of Automotive Turbochargers, DOI: 10.1007/978-3-642-35070-2_3, Ó Springer-Verlag Berlin Heidelberg 2013

15

16

3 Acoustic Propagation Theory

• Oil films of bearing • Bearing center housing • Compressor housing • Charge- air intercoolers • Air filter • Exhaust-gas system (Catalyzer, DPF, & Muffler)

Airborne noise Car frame

Fig. 3.1 Transmitting ways of airborne noise

pressure and density in a period of time T, as shown in Fig. 3.2. At v = 0, the oscillating air particle reaches the maximum amplitude where the air pressure and density equal the time-averaged or undisturbed air pressure and density (p0, q0). The compression and expansion waves take turns changing in the wavelength k in the propagation direction. After the compression wave, the expansion wave follows in a period of time T and vice versa. While the air particles oscillate with the velocity v(t) about their equilibriums, the perturbed noise pressure p0 moves with the sound velocity c in all propagation directions. However, both velocities v and c are in the same direction of the wave propagation direction. p ' ( x0 , t0 )

ξ ( x, t )

ν ( x, t ) =

∂ξ ( x, t ) ∂t

p ' ( xn , t n )

c

c

(sound speed)

v

p0 , ρ 0

t

compression wave (v > 0)

expansion wave (v < 0)

compression wave (v > 0)

T 0 Fig. 3.2 Acoustic propagation wave characteristics

x

λ = cT

3.1 Aeroacoustic Characteristics

17

Fig. 3.3 Noise wave characteristics at a given position x

Some thermodynamic characteristics of noise transmitting in a compressible fluid are necessary for noise pressure propagation in the linear aeroacoustics (see Appendix A). The linear aeroacoustics is based on the linearized Euler’s equations (LEE) for inviscid and isentropic flows in which the acoustic propagation characteristics of p0 , q0 , and T0 have relatively small perturbation amplitudes in the fluid. (a) Speed of sound The sound velocity c is generated from molecular interactions between the oscillating air particles with the fluctuation velocity v about the equilibriums and only depends on the absolute fluid temperature of the transmitting medium. The speed of sound c is defined by the square root of the change rate of the thermodynamic absolute fluid pressure p to its density q in an isentropic process. sffiffiffiffiffiffiffiffiffiffiffiffiffiffi   op c¼ ð3:1aÞ oq s The acoustic characteristics p0 , q0 , and T0 are defined as (see Fig. 3.3). p ¼ p0 þ p0 ðx; tÞ;

q ¼ q0 þ q0 ðx; tÞ;

T ¼ T0 þ T 0 ðx; tÞ

where p0 p0 (x, t) q0 q0 (x, t) T0 T0 (x, t)

ð3:1bÞ

is the time-averaged or undisturbed gas pressure (v = 0); is the perturbed noise pressure (p0 p0); for convenience only, it is further called the noise pressure p0 ; is the time-averaged or undisturbed gas density (v = 0); is the perturbed gas density (q0  q0); is the time-averaged or undisturbed gas temperature (v = 0); is the perturbed gas temperature (T0  T0) thus, T & T0.

18

3 Acoustic Propagation Theory

The relation of pressure and density of a perfect gas is described in an isentropic process. p ¼ Cqj

ð3:2Þ

where C is an integral constant; j is the isentropic coefficient of gas (rair = 1.4). The state equation of a perfect gas gives the relationship between pressure, gas density, and temperature. p ¼ qRT

ð3:3Þ

where R is the gas constant; q is the gas density; T is the absolute gas temperature [K]. Having substituted Eqs. (3.2 and 3.3) into Eq. (3.1a), one obtains the sound speed in a compressible fluid of a perfect gas (e.g. air) at T & T0. rffiffiffiffiffiffi jp pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ¼ jRT  jRT0 ¼ c0 ð3:4Þ c¼ q

Equation (3.4) gives the sound speed in dry air c0 & 340 ms-1 at T0 = 293.14 K (20 °C). The longitudinal sound speed in incompressible media (e.g. solid materials and liquids) is calculated by [2] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eð1  mÞ K E cs ¼ ; cl ¼ ð3:5Þ ¼ qð1 þ mÞð1  2mÞ q 3qð1  2mÞ within E is v is K is q is

the the the the

medium medium medium medium

elasticity modulus (Young’s modulus); Poisson’s ratio (v = 0.25 to 0.30 for most metals); bulk modulus; density.

As a result, the sound speed cs is nearly 6,000 m/s in steel: cl is about 1,500 m/s in liquid water at 25 °C according to Eq. (3.5). The relation between the noise pressure p0 , perturbed gas density q0 , and perturbed gas temperature T 0 is derived from Eq. (3.1b) and the state equation of a perfect gas, as given in Eq. (3.3).

3.1 Aeroacoustic Characteristics

19

   q0 T0 1þ : 1þ q0 T0 p0 q0 T 0 p0 q0 T 0 ¼ þ ,1þ 1þ þ ) p0 q0 T0 p0 q0 T0

p q T ¼ : , p0 q0 T0



1þ

p0 p0



¼

ð3:6Þ

Using Eq. (3.2), it can be written for a perfect gas at p0  p0 and q0  q0 in an isentropic process.  0    0 p q0 j q 1þ 1þj ¼ 1þ p0 q0 q  0  00  ð3:7aÞ p q ) j p0 q0 Therefore, one obtains the relation between the perturbation noise pressure and density according to Eqs. (3.4 and 3.7a).   p0 p0  j ð3:7bÞ ¼ jRT0 ¼ c20 0 q0 q Thus, the perturbed noise pressure is proportional to the perturbed density at very small perturbation amplitudes with c0 & c in linear aeroacoustics. p0 ¼ c20 q0  c2 q0

ð3:8Þ

The perturbed gas temperature is calculated from Eqs (3.6 and 3.7a). T0 p0 q0 ¼  T0 p0 q0   q0 j  1 p0 ¼ ðj  1Þ  j q0 p0

ð3:9Þ

According to Eq. (3.9), the perturbed air temperature T0 & 0.57.10-4 T0 and perturbed air density q0 & 1.43.10-4 q0 at a perturbed noise pressure p0 = 20 N/ m2 and p0 = 105 N/m2. This result shows that T0  T0 and q0  q0, as given in Eq. (3.1b). (b) Noise impedance The impulse equation of an air particle oscillating with an amplitude n at a velocity v is written in the direction x. q0

o2 n op0 ¼ 2 ot ox

ð3:10Þ

Thus, o2 n om 1 of ðt  x=cÞ ¼ : ¼ ot2 ot q0 ox

ð3:11Þ

20

3 Acoustic Propagation Theory

The forward propagating noise pressure can be written in the direction x. p0 ðx; tÞ ¼ f ðt  x=cÞ  f ðsÞ within the delayed time is defined by s  ðt  x=cÞ: Having timely integrated both sides of Eq. (3.11), one obtains the propagating noise velocity v using the differentiation chain rule. Z Z 1 of ðsÞ os 1 of ðsÞ ot : dt ¼ : dt v¼ q0 os ox q0 c ot os Z Z 1 op0 ðx; tÞ ot 1 op0 ðx; tÞ ¼ : dt ¼ dt ð3:12Þ q0 c ot os q0 c ot p0 ¼ q0 c where os 1 os ¼ ; ¼1 ox c ot Noise impedance Z is defined by the ratio between the noise pressure and air particle velocity; it is resulted from Eq. (3.12). Z  q0 c ¼

p0 ðx; tÞ mðx; tÞ

ð3:13Þ

According to Eq. (3.13), the noise impedance (q0c) in air is about 428 kg/(m2s) at 0 °C (273.14 K); approximately 415 kg/(m2s) at 20 °C (293.14 K). Thus, the air particle velocity v is approximately 0.05 m/s at a noise pressure p0 = 20 N/m2 (Pa). (c) Noise intensity Noise intensity vector I is the product of the noise pressure and air particle velocity. Iðx; tÞ ¼ p0 ðx; tÞ vðx; tÞ where p0 (x,t) v(x,t)

ð3:14aÞ

is the noise pressure, as given in Eq. (3.1b); is the air particle velocity.

The time-averaged noise intensity vector is given over the time period T. 1 Imean ðxÞ ¼ T

ZT 0

p0 ðx; tÞvðx; tÞdt

ð3:14bÞ

3.1 Aeroacoustic Characteristics

21

The rms noise intensity amplitude is calculated from Eqs. (3.13 and 3.14a). 2

Irms ðxÞ ¼

2

p0rms ðxÞ p0rms ðxÞ ¼ Z q0 c

ð3:15Þ

where p0 rms is the root mean square of the noise pressure vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ZT u u1 p0rms ðxÞ ¼ t p02 ðx; tÞdt T

ð3:16Þ

0

(d) Noise power Noise power is defined by the sum of the noise intensity over the surface S surrounding the noise source. I P ¼ I:n dS ð3:17Þ S

The time-averaged noise power is resulted from the time-averaged noise intensity and surrounding surface.  ¼ IS ¼ S P T

ZT

IðtÞ  n dt

ð3:18Þ

0

(e) Noise levels There are two kinds of noise levels in the aeroacoustics: noise power level and pressure level. • Noise power level Noise power level LW [dB] is defined by LW ¼ 10 log10



 P Pref



ð3:19Þ

within  is the time-averaged noise power [W], as given in Eq. (3.19); P Pref is the reference noise power (=10-12 W). The time-averaged noise power is proportional to the noise intensity according to Eqs. (3.15 and 3.18); therefore, one can write in

22

3 Acoustic Propagation Theory

P / Irms / p02 rms

ð3:20Þ

• Noise pressure level Noise pressure level LP [dB] is defined by  0 2  0    Irms prms p ¼ 20 log10 rms ¼ 10 log10 LP ¼ 10 log10 pref pref Iref

ð3:21aÞ

where p0 rms is the rms noise pressure [N/m2], as given in Eq. (3.16); pref is the reference noise pressure (= 2.10-5 N/m2). According to Eq. (3.21a), the noise pressure level is about 114 dB at an effective pressure p0 rms = 10 N/m2 (Pa); LP = 120 dB at p0 rms = 20 N/m2 (Pa). As a result, the noise pressure level increases by nearly 6 dB if the rms noise pressure is doubled. Obviously, the difference in noise pressure level depends on the rms noise pressure ratios. ! p02;rms DLP ¼ LP;2  LP;1 ¼ 20 log10 0 ð3:21bÞ p1;rms

3.2 Acoustic Propagation Equations Vibrating solid surface causes local propagating disturbances, such as timely change rates of pressure, density, and temperature in the fluid at the solid surface. The gaseous fluid molecules interact with each other and radiate noise from the noise source to the transmitting medium. Note that the larger the timely change rate of the gas density or pressure, the more effective the noise propagates with radiation velocity v that is very close to the average molecular gas velocity in the gas medium [1]. Acoustic propagation is caused by the fluid molecular interaction due to the initial disturbances at the interface between the solid surface and fluid. The noise propagation is based on the continuity and momentum equations in the gaseous medium that is assumed as a perfect gas (e.g. air). The conservation of mass (continuity) equation of gas with the absolute gas density q is written in oq þ r:ðqvÞ ¼ 0 ot

,

oðq0 þ q0 Þ þ r:ðq0 v þ q0 vÞ ¼ 0 ot

ð3:22aÞ

3.2 Acoustic Propagation Equations

23 0

0 According to Fig. 3.3, it gives oqot0 \\ oq ot , and r:ðq vÞ \\ r:ðq0 vÞ Thus, the continuity equation in Eq. (3.22a) becomes

oq0 þ r:ðq0 vÞ ¼ 0 ot where q q0 q0 v r:ðq0 vÞ

is is is is is

the the the the the

ð3:22bÞ

absolute gas density given in Eq. (3.1b); perturbed gas density; undisturbed gas density; fluctuating fluid velocity in the transmitting medium; divergence of the vector q0 v:

The Nabla operator r is defined in • Cartesian coordinate system (x,y,z):

r¼

o o o iþ jþ k ox oy oz

ð3:23aÞ

• Spherical coordinate system (r,h,u):

r¼

o 1 o 1 o er þ : eh þ : e/ or r oh r sin h o/

ð3:23bÞ

Because q0 \\q0, v is very small, and p0 unchanged with locations, it results in o(qvÞ o(q0 vÞ  ot ot p0 (x; y; z) ¼ const ) rp ¼ rp0 q0 v\\q0 v )

The conservation of momentum equation of viscous gas flows is given by oðq0 vÞ þ r:ðq0 v  vÞ ¼ rp0 þ r:s0 ot where rp0 r.s0 vv

ð3:24Þ

is the noise pressure vector acting on the fluid; is the perturbed viscous shear tensor; is the tensor product of the gas velocity.

The relation of the timely change rates of propagating noise pressure and perturbed gas density in an isentropic process is resulted from Eq. (3.8).

24

3 Acoustic Propagation Theory

op0 oq0 ¼ c20 ot ot

ð3:25Þ

Having combined Eqs. (3.22b and 3.25), it results in op0 þ c20 r:ðq0 vÞ ¼ 0 ot

ð3:26Þ

Timely differentiating Eq. (3.26), one achieves   o2 p0 o2 p0 oðq0 vÞ 2 o 2 r:ðq þ c þ c r: vÞ ¼ ¼0 0 0 0 ot ot ot2 ot2

ð3:27Þ

The convective term, the second term on the left-hand side of Eq. (3.24), is second order of the gas velocity v and therefore mostly negligible in linear aeroacoustics due to small oscillating air-particle velocities according to Eq. (3.13). Thus, Eq. (3.24) becomes for inviscid gas flows oðq0 vÞ  rp0 ot

ð3:28Þ

Substituting Eq. (3.28) into (3.27), one obtains the acoustic propagation equation linearized Euler’s equation (LEE), in which the viscous shear in the gas flow is negligibly small, and the perturbation amplitudes are very low. Linear acoustics generally applies the LEE, derived from the Navier-Stokes equations at low Reynolds numbers of perfect gases in an isentropic process, to study the noise propagation in the transmitting fluid. o2 p0 o2 p0 2 2 0  c r p   c20 Dp0 ¼ 0 0 ot2 ot2

ð3:29aÞ

where p0 is the perturbed noise pressure; c0 is the time-averaged or undisturbed sound speed, as given in Eqs (3.1a, 3.4, and 3.5); Dp0 is the Laplacian operator of the noise pressure p0 . The wave equation of a scalar velocity component v(x,y,z,t) is given by substituting Eq. (3.13) into Eq. (3.29a). r2 v 

1 o2 v 1 o2 v  Dv  ¼0 : : c20 ot2 c20 ot2

The Laplacian operator Du with u = p0 ; v(x,y,z,t) is formulated in

ð3:29bÞ

3.2 Acoustic Propagation Equations

25

• Cartesian coordinate system (x,y,z):

Du  r2 u ¼ r:ru ¼

o2 u o2 u o2 u þ þ 2 ox2 oy2 oz

ð3:30aÞ

• Spherical coordinate system (r,h,/) in the radial direction r:

Du  r2 u ¼ r:ru ¼

  1 o o2 u 2 ou 2 ou r : þ : ¼ r 2 or or or 2 r or

ð3:30bÞ

Similarly, the Lighthill’s equation (also Lighthill’s analogy) is derived from the nonlinear Navier-Stokes equations of viscous gas flows at high Reynolds numbers and aeroacoustics. It has been considered as the bridge between aerodynamics and aeroacoustics for the acoustic field. The Lighthill’s equation describes the acoustic propagation at large perturbation amplitudes of viscous gas flows with high Reynolds numbers in nonlinear aeroacoustics due to the nonlinear terms on the rhs [2]. 8 2 0 o q > > < 2  Dp0 ¼ r:r:ðq0 v  v  s0 Þ , ot ð3:31aÞ 2 0 2 > o p o > 0 2 2 0 0 2 0 :  c0 Dp ¼ c0 r:r:ðq0 v  v  s Þ þ 2 ðp  c0 q Þ ot ot2

within q0vv s0

is the convective tensor; is the perturbed viscous shear tensor.

The non-isentropic source term of q2/qt2(p0 - c02q0 ) on the rhs of Eq. (3.31a) describes the nonlinear equivalent acoustic source in the Lighthill’s equation. However, this term vanishes at small perturbation amplitudes in the isentropic noise propagation.   Having substituted Dp0 ¼ c20 Dq0 þ r:r: ðp0  c20 q0 ÞI into the first equation of (3.31a), the Lighthill’s equation is given in the perturbed gas density. o2 q0  c20 Dq0 ¼ r:r:T ot2

ð3:31bÞ

The Lighthill’s turbulence tensor T is given by T  q0 v  v  s0 þ ðp0  c20 q0 ÞI where I is the unity tensor equivalent to the Kronecker delta tensor.

ð3:31cÞ

26

3 Acoustic Propagation Theory

In the case of inviscid flows at low Reynolds numbers and small perturbation amplitudes in an isentropic process for linear acoustics, all rhs terms in Eqs. (3.31a and 3.31b) equal zero. Thus, the Lighthill’s equation (3.31a) in the perturbed pressure becomes the LEE given in Eq. (3.29a). The acoustic propagation equation LEE of the average noise pressure p0 is written in the radial direction r.   0 o2 p0 c20 o 2 op r  ¼0 ð3:32Þ ot2 or r 2 or Substituting a new unknown variable m = p0 r into Eq. (3.32), one obtains the acoustic propagation equation LEE in the radial direction after a few arithmetic calculations. o2 ðmÞ o2 ðmÞ  c20 ¼0 2 ot or 2

ð3:33Þ

The noise pressure in Eq. (3.33) can be written for the forward and backward propagating waves in the complex formulation m ¼ p0fw ðr; tÞ þ p0bw ðr; tÞ r A1 A2 ¼ ejxðtr=cÞ þ ejxðtþr=cÞ r r

p0 ðr; tÞ ¼

ð3:34Þ

where Ai is the integrating coefficients; x is the oscillation angular frequency; r is the noise transmitting distance. The wavenumber k is defined by the ratio of the angular frequency of air particles x to time-averaged or undisturbed sound speed c0. k 

x x 2p ¼  c0 c k

ð3:35Þ

where k is the wavelength of acoustic propagation with k = cT. The noise pressure is generated by the spherical surface that vibrates in the radial direction r with a velocity v = xn at frequency x about its equilibrium radius a. The complex noise pressure in the forward propagating direction is resulted in [1]. p0 ðr; tÞ ¼

1 q jx Q jkðraÞ : 0 e ð1 þ jkaÞ 4pr

ð3:36Þ

within the complex volume velocity of the noise source Q [m3/s] is written in Q ¼ Q ejxt ¼ ð4pa2 v)ejxt

ð3:37Þ

3.2 Acoustic Propagation Equations

27

The noise source strength is defined by q0jxQ* which is given in Eq. (3.38). _ ¼ q0 q0 Q

d  jxt ðQ e Þ ¼ q0 jx Q ejxt ¼ q0 jx Q dt

ð3:38Þ

The complex volume acceleration dQ/dt [m3/s2] causes the noise propagation at the velocity v by sound radiation in the transmitting medium. For a point monopole source (i.e., ka  1), the noise pressure is given by substituting Eq. (3.38) into Eq. (3.36). _ q0 jx Q jkr q0 Q e ejkr ¼ 4pr 4pr q jx Q jðxtkrÞ e ¼ 0 4pr

p0 ðr; tÞ ¼

ð3:39Þ

The inverse square law of the noise intensity is resulted from calculating the root mean square of p0 (r,t). 2

Irms / p0rms ¼

ðq0 ckÞ2 2 Q 32p2 r 2

ð3:40Þ

This inverse square law indicates that the noise intensity decreases with the square transmitting radius r; i.e., the longer the distance from the noise source, the lower the noise level is generated according to Eq. (3.40).

3.3 Computation of Acoustic Propagations 3.3.1 Computational Methods The noise computational simulation of automotive turbochargers is much more complex and difficult than computational fluid dynamics (CFD) because the airborne noise computation must be divided into many interdisciplinary simulations based on finite element method (FEM), finite volume method (FVM), and boundary element method (BEM), such as the noise generation simulations using CFD and computational nonlinear rotordynamics (CNR), in the near-field acoustic propagation simulations using computational aeroacoustics including LEE for linear aeroacoustics, acoustic perturbation equations (APE) using the Lighthill’s equation for nonlinear aeroacoustics in the mid-field, and far-field simulations of noise radiation into the environment using the Ffowcs Williams and Hawkings’ equation (FWHE) based on the integral method BEM, as shown in Fig. 3.4. The BEM solves the Kirchhoff-Helmholtz integral equation for 3-D acoustic field problems by summation of the noise pressures over the boundary surface of the noise source body.

28

3 Acoustic Propagation Theory r

Near-field

CFD, CRD wavefronts

t Mid-field

p´ (r,t)

compression waves expansion waves

CAA: LEE, APE

Far-field

BEM: FWHE

Fig. 3.4 Acoustic propagation computational methods

First, the generation of aerodynamic noise sources, such as flow vortexes and turbulent flows, is transiently computed by CFD with a direct numerical simulation (DNS) for which all flow initial and boundary conditions, and turbocharger geometries must be known in advance. CNR is applied to simulate the rotordynamic noise sources in the turbocharger (see Chap. 5). Second, using the LEE and APE the acoustic propagation through the components neighboring turbochargers are computed under mutual influences of the component walls on the noise levels. Third, the FWHE is used to compute the noise radiation neglecting the component wall effects at different rotor speeds in the far-field of the turbocharger (i.e. within a few meters from the vehicle). The Lighthill’s equations given in Eqs. (3.31a and 3.31b) with the noise source terms on the rhs are usually solved by the three-dimensional CAA program (3-D CAA) based on the FEM and BEM for complex systems [1–3]. As an example, such complex simulation models are applied in automotive industries where the noise sources induced from the turbocharger, as discussed in Sect. 3.2, are transmitted through the bearing center housing, compressor housing, charge-air intercoolers, air inlet system, exhaust-gas system (catalyzer, DPF, and muffler), and car frame to the cabin. Additionally, the strengths of aero- and rotordynamic noise sources Qi on the rhs term of the Lighthill’s equation are computed by 3-D CFD of flows in the compressor and turbine wheels [4] and the CNR of the turbocharger [5, 6], respectively. The BEM and FEM are briefly outlined for linear aeroacoustics in the following subsections.

3.3 Computation of Acoustic Propagations Fig. 3.5 BEM in an exterior acoustic field (Va)

29 →

n

(S∝ )

p' (ra , t ) ra x3

p' (r, t )

(Sa)

(Va)

→

n

r (S0)

0

acoustic volume domain

noise radiating surface

v(ra , t ) ∆Sj

x2

x1

turbocharger

3.3.2 BEM for Aeroacoustics The BEM based on Green’s second theorem is used to transform the Helmholtz equation in the acoustic volume domain (Va) into the Kirchhoff-Helmholtz integral equation defined on the closed surface (Sa). The noise propagating domain (Va) is enclosed by the closed surfaces S?, S0, and Sa, as shown in Fig. 3.5. The noise pressure p0 (r,t) at any point r in the noise field (Va) is solved by the Kirchhoff-Helmholtz integral equation at the given boundary conditions of p0 (ra,t) and v(ra,t) on the boundary surface (Sa). The homogeneous noise pressure equation given in Eq. (3.29a) is written in the acoustic field volume domain. r 2 p0 

1 o2 p0 ¼0 c20 ot2

ð3:41Þ

The timely second derivative of the harmonic noise pressure p0 can be written in p0 ðr; tÞ ¼ ^ p0 ðrÞ ejxt )

o2 p0 ðr; tÞ ¼ x2 p0 ðr; tÞ ot2

ð3:42Þ

The homogeneous Helmholtz equation of the noise pressure p0 is resulted from Eqs. (3.35, 3.41, and 3.42). r2 p0 ðr; tÞ þ k2 p0 ðr; tÞ ¼ 0; 8r 2 Va k  x=c0  2p=k

ð3:43Þ

In the case of a noise source Q(r,t) in the acoustic field Va, the inhomogeneous Helmholtz equation is formulated in r2 p0 ðr; tÞ þ k2 p0 ðr; tÞ ¼ Qðr; tÞ; 8r; Q 2 Va where the noise source in the rhs is defined by [1]

ð3:44aÞ

30

3 Acoustic Propagation Theory

Qðr; tÞ ¼ r:fðr; tÞ  jq0 x qðr; tÞ

ð3:44bÞ

The rhs term in Eq. (3.44a) describes the force and noise source intensity acting on the acoustic field volume domain, as described in the Lighthill’s equation (3.31b and 3.31c) according to [1]. Generally, there are three kinds of boundary conditions at the enclosed surface (S) for the Helmholtz equation: • Dirichlet boundary condition at the rigid surface

op0 ¼ 0 ) p0ðSÞ ðra ; tÞ ¼ constant on ðSÞ

ð3:44cÞ

• Neumann boundary condition at the vibrating surface

op0 ¼ jq0 x vðSÞ ðra ; tÞ on ðSÞ

ð3:44dÞ

• Robin boundary condition (also impedance BC for Sturm-Liouville problems)

p0ðSÞ ðra ; tÞ op0 q x ; Z  q0 c  0 ¼ jq x 0 k on ðSÞ Z

ð3:44eÞ

where Z is the noise impedance of the acoustic field, as given in Eq. (3.13). Green’s second theorem transforms the volume domain V into the boundary S. I Z ow ou  w ÞdS ¼ ður2 w wr2 uÞdV ðu ð3:45Þ on on S

V

where u and w are two scalar functions defined in the volume domain V. To apply the BEM to linear aeroacoustics, the noise pressure p0 (ra,t) at the location ra in the acoustic field Va is substituted for u; the free-field Green’s function G(r,ra) in the acoustic field Va with the wavenumber k is used for w [1, 7]. ( /  p0 ðra ; tÞ; ð3:46Þ ejkjrra j x 2p w  Gðr; ra Þ ¼ 4p jrra j ; k  c0  k Having applied the Green’s second theorem to the closed boundary surfaces of S?, S0, and Sa, one obtains the Kirchhoff-Helmholtz integral equation by setting the rhs term in Eq. (3.45) to equal zero because both p0 (ra,t) and G(r,ra) satisfy the homogeneous Helmholtz equation (3.43).

3.3 Computation of Acoustic Propagations

p0 ðr; tÞ ¼

1 cðrÞ

31

 I  oGðr; ra Þ op0 ðra ; tÞ  Gðr; ra Þ: p0 ðra ; tÞ: dSa on on

ð3:47aÞ

Sa

where the domain coefficient c(r) is resulted in [1] 8 1 if 8r 2 Va > > < 1 cðrÞ ¼  if 8r 2 Sa 2 > > : 0 if 8r 62 Va

ð3:47bÞ

The vibrating velocity of the solid surface is given by the Neumann boundary condition given in Eq. (3.44d). op0 ðra ; tÞ ¼ jq0 x vðSa Þ  jq0 ck vðSa Þ on ðSa Þ ð3:48Þ j op0 ðra ; tÞ ) vðra ; tÞjðSa Þ ¼ : on q ck 0

ðSa Þ

Integrating the rhs term of Eq. (3.47a) at the noise pressure p0 (ra,t) and vibrating velocity v(ra,t) on the boundary surface Sa, it gives the noise pressure at any point r in the noise field Va. In the case of r inside Va at c(r) = -1, the induced noise pressure results in  I  op0 ðra ; tÞ oGðr; ra Þ  p0 ðra ; tÞ: ð3:49Þ Gðr; ra Þ: p0 ðr; tÞ ¼ dSa on on Sa

The noise pressure p0 is written in the Helmholtz operators by substituting the additional incident acoustic field p0in into Eq. (3.49). p0 ðr; tÞ ¼ fLvgSa fMp0 gSa þp0in ðr; tÞ; 8r 2 Va

ð3:50aÞ

in which the incident noise pressure p0in is resulted from integrating the source term Q(r,t) over the acoustic field volume VS. I 0 pin ðr; tÞ ¼ Qðrs ; tÞ:Gðr; rs Þ dVS ð3:50bÞ VS

The Helmholtz integral operators on the boundary surface Sa are defined by 8 i H h op0 ðra ;tÞ > Lv  Gðr; r Þ: dSa ; > f g a Sa < on Sa h i ð3:51Þ H 0 aÞ 0 > > p ðra ; tÞ: oGðr;r dSa : fMp gSa  on Sa

The integrated boundary surface Sa is divided into N finite boundary elements DSj=1,…,N, as shown in Fig. 3.5. Thus, the noise pressure of Eq. (3.50a) inside the

32

3 Acoustic Propagation Theory

acoustic field Va is resulted by integrating the rhs of Eq. (3.50a) over the finite boundary surfaces DSj=1,…,N. p0 ðr; tÞ ¼

N X j¼1

fLk egDSj vj ðra ; tÞ 

N X j¼1

fMk egDSj p0j ðra ; tÞ þ p0in ðr; tÞ; 8r 2 Va

where e is the unit vector on the finite boundary element DSj. The Helmholtz integral operators in Eq. (3.52) are given by R ( fLk egDSj Sa  DSj Gk ðr; ra ÞdSa ; R ðr;ra Þ dSa fMk egDSj Sa  DSj oGkon

ð3:52Þ

ð3:53Þ

Due to the solution singularity for certain values of the wavenumber k on the surface domain, other integrating methods, such as Schenck method based on combined Helmholtz Integral equation formulation (CHIEF), improved indirect, and direct formulations are used to compute the noise pressure in the acoustic field [1, 2, 7]. The commercial 3-D Acoustics Code of LMS Virtual Lab [8] based on the boundary element acoustics (BEA) is mostly used to compute three-dimensional acoustic field problems in automotive applications.

3.3.3 FEM for Aeroacoustics Instead of solving the Helmholtz equation by integrating the noise pressure over the radiating surface (Sa) by means of BEM, the FEM directly solves this equation in a three-dimensional acoustic filed (V) surrounding the radiating surface of the turbocharger, as shown in Fig. 3.6. Multiplying the Helmholtz equation by a scalar function / differentiable in the control volume V and integrating both sides of the equation over dV, one achieves the weak Galerkin formulation of the Helmholtz equation: Fig. 3.6 FEM in an acoustic field (V)

3.3 Computation of Acoustic Propagations

Z

0

33

/ r:rp dV þ

V

Z

/ k2 p0 dV ¼ 0

ð3:54Þ

V

Having used the product chain rule of the divergence r:ð/:rp0 Þ ¼ /:r:rp0 þ r/:rp0 ; Equation (3.54) is rewritten in Z Z Z r:ð/rp0 ÞdV  r/:rp0 dV þ / k2 p0 dV ¼ 0 V

V

ð3:55Þ

V

Applying Gauss divergence theorem to the control volume V, it is given by Z I 0 r:ð/rp ÞdV ¼ ð/rp0 Þ: n dS V

S

¼

I

/

op0 dS ¼ jq0 x vSa on

Z

/ dS

Sa

S

Substituting Eq. (3.56) into Eq. (3.55), one obtains Z Z 0 2 0 ðr/:rp  k / p ÞdV ¼ jq0 x vSa / dS V

ð3:56Þ

ð3:57Þ

Sa

within the fluid velocity vSa on the radiating surface satisfies the Neumann boundary condition given in Eq. (3.44d). j op0 : vSa ¼ ð3:58Þ q0 x on ðSa Þ The Rayleigh-Ritz method approaches the noise pressure p0 (e) in the finite element e = 1,…,Ne in the local coordinate system (n1, n2, n3) by a finite series expansion according to [1].



  p0  p0 ðeÞ  Sðn1 ; n2; n3 Þ :fag ¼ Sðn1 ; n2; n3 Þ : ½De 1 :fp0 g N X

 wj ðn1 ; n2; n3 Þp0j ¼ wðn1 ; n2; n3 Þ :fp0 g  j¼1

where S is a is De is w is p0 is

the the the the the

shape function (1 9 N) matrix; coefficient (N 9 1) vector; distribution function (N 9 N) matrix; weighting function (1 9 N) matrix; noise pressure (N 9 1) vector;

ð3:59Þ

34

3 Acoustic Propagation Theory

Ne N wj

is the number of discretized finite elements in the control volume V; is the nodal points of the discretized finite element; is the pressure weighting function at the nodal point j that denotes the weighting rate of the noise pressure per unit volume; is the noise pressure at the nodal point j.

p0j

Inserting Eq. (3.59) into Eq. (3.57), it gives at choosing / : wi. 2 3 Z Z N X 4 ðrwi :rwj  k2 wi wj Þ dV 5:p0 ¼ jq0 x vSa wi dS; 8i ¼ 1; . . .; N j j¼1

V

Sa

ð3:60Þ

Thus, Eq. (3.60) is formulated in the matrix equation of the noise pressure. ½A : fp0 g ¼ fqg where [A] = (N 9 N)

aij 

Z

ð3:61aÞ

matrix of the components

ðrwi :rwj  k2 wi wj Þ dV; 8i ¼ 1; 2; . . .; N; 8j ¼ 1; 2; . . .; N

ð3:61bÞ

V

0

{p } = (N x 1)

{q} = (N 9 1)

vector of the noise pressures  T fp0 g ¼ p01 p0i . . . p0N ; 8i ¼ 1; 2; . . .; N

vector of [q1 q2

…

qi ¼ jq0 x vSa

ð3:61cÞ

qN]T of the components

Z

wi dS; 8i ¼ 1; 2; . . .; N

ð3:61dÞ

Sa

The weighting functions w of the considered finite element depend on the discretizing scheme [1, 7]: • For a local orthogonal curvilinear coordinate (n1, n2, n3), the weighting functions wj (n1, n2, n3) based on a quadratic element with N = 4 nodal points are given

fw4NP ðn1 ; n2 ; n3 Þg ¼ ½ w1

w2

w3

w4

ð3:62Þ

3.3 Computation of Acoustic Propagations

35

• For a local orthogonal curvilinear coordinate (n1, n2, n3), the weighting functions wj (n1, n2, n3) based on a hexahedral element with N = 8 nodal points are given

fw8NP ðn1 ; n2 ; n3 Þg ¼ ½ w1

w2

w3

w4

w5

w6

w7

w8

ð3:63Þ

To transform the integral Eqs. (3.60, 3.61b and 3.61d) from the orthogonal curvilinear coordinate (n1, n2, n3) of the finite element mesh into the Cartesian coordinate (x1, x2, x3), the Jacobian determinant J is used to compute the gradients of w and dV in the curvilinear coordinate (see Appendix B). The commercial 3-D Acoustics Code ACTRAN Aero and Vibroacoustics of the MSC Software [10] based on the FEM is mostly used to compute three-dimensional acoustic field problems in automotive applications. The following computing steps must be carried out to find the noise pressure solution of Eq. (3.61a) in the acoustic field V: • Compute the terms aij in Eq. (3.61b) and the rhs term qi in Eq. (3.61d); • Compute the noise pressures p0j at the nodal points by solving the matrix equation (3.61a) at the boundary conditions given in Eqs. (3.44c–e); • Compute the mean noise pressure in the discretized finite elements according to Eq. (3.59).

3.4 Frequency Modulations of Nonlinear Aeroacoustics Nonlinear aeroacoustics deals with large noise perturbation amplitudes in which the acoustic characteristics are no longer linear. Therefore, synchronous and asynchronous responses occur in a nonlinear aeroacoustic system when first, the aero- and rotordynamic noise responses are superimposed to generate sidebands and; second, the airborne noise response is additionally generated by superimposing many acoustic propagation waves at different frequencies during propagation in the far-field at high Reynolds numbers. The sidebands occurring at the sub- and supersynchronous frequencies are generated by the propagated noise waves at different frequencies. Note that the more visibly the sidebands appear in the sound spectra plot (Waterfall plot), the more nonlinearly the aeroacoustic system responds. The response amplitude of the nonlinear aeroacoustic system is resulted from all component amplitudes of the propagating noise waves and sidebands combined together. As a result, the noise response amplitude is intensified; the aeroacoustic behavior becomes much more nonlinear at large perturbation amplitudes in turbulent flows at high Reynolds numbers.

36

3 Acoustic Propagation Theory

3.4.1 Responses of Nonlinear Aeroacoustic Systems Having considered a nonlinear aeroacoustic system, the system response y(x) is resulted from the input signal x(t) with large amplitudes and the system transfer impedance Z(e), as shown in Fig. 3.7. The nonlinear response y(x) of the system is written in second order of the input function x(t).   yðxÞ ¼ f ðx; eÞ ¼ Z x þ e:sgnðxÞx2 þ . . . ð3:64Þ ¼ Zðx þ e x2 þ . . .Þ where x Z Zx e Ze*x2 e* = e.sgn(x)

is the input signal with large amplitudes; is the complex transfer impedance of the system; is the linear term of the response; is the nonlinear system factor (e  1); is the nonlinear term of the response; within sgn(x) is the signum function of x, which is defined by

Therefore,

8 < 1 if x\0 sgnðxÞ ¼ 0 if x ¼ 0 : þ1 if x [ 0 8 < e if x\0 e  esgnðxÞ ¼ 0 if x ¼ 0 : þe if x [ 0

Thus, the nonlinear vibration response becomes

yðxÞ ¼ Zx þ ZesgnðxÞx2 þ Oðx3 Þ

ð3:65Þ

The nonlinear vibration response y(x) is displayed in Fig. 3.8 and compared to the linear response. In the linear system, the response of Zx is proportional to the input function x. On the contrary, the nonlinear response is different from the linear term of Zx; it increases or decreases more quickly with the input amplitude x(t) because the nonlinear term is proportional to x2. As a result, the nonlinear system can only be linearized at the small amplitudes because the nonlinear term is quite small. Fig. 3.7 Response of a nonlinear aeroacoustic system

Input

Nonlinear system

x(t )

Z (ε )

Output

y ( x) = f ( Z , ε , x)

3.4 Frequency Modulations of Nonlinear Aeroacoustics

37

Fig. 3.8 Comparison of the linear and nonlinear responses

y nonlinear linear

Z (x + ε *x2 )

Zε * x 2

Zx

(nonlinear term)

0

x

x

In case of the excessively large unbalance whistle amplitude, the system behavior is strongly nonlinear. Thus, the response amplitude y(x) increases faster than the amplitude of the linear system since the nonlinear term increases with the input amplitude squared that is no longer negligible, as shown in Fig. 3.8.

3.4.2 Modulated Noise Sideband Frequencies Having supposed that the input x(t) is a harmonic acoustic propagation wave with the amplitude X and single frequency x1. x ¼ X cos x1 t

ð3:66Þ

Its response is resulted from Eq. (3.64) in second order of x(t). y ¼ ZX cos x1 t þ Ze X 2 cos2 x1 t  yLterm þ yNLterm

ð3:67Þ

The noise response consists of two terms, the first term yL-term is linear with x(t) and the latter yNL-term is the nonlinear term of x2. The nonlinear term can be written in another trigonometric formulation. yLterm  ZX cos x1 t;

ð3:68Þ

yNLterm  Ze x2 ¼ Ze X 2 cos2 x1 t 1 ¼ Ze X 2 ð1 þ cos 2x1 tÞ 2

ð3:69Þ

38

3 Acoustic Propagation Theory

By substitution of Eqs. (3.67, 3.68, and 3.69), one gets the overall noise response. 1 1 y ¼ ZX cos x1 t þ Ze X 2 þ Ze X 2 cos 2x1 t 2 2

ð3:70Þ

Compared to the linear system, the nonlinear response has two new additional terms: the rectification term of X2 and the second-order harmonic term of 2x1. Now, we consider the input function x(t) of a periodic noise containing two frequency components of x1 and x2 and corresponding amplitudes X1 and X2. x ¼ X1 cos x1 t þ X2 cos x2 t; x1 [ x2

ð3:71Þ

To simplify, we consider only the nonlinear term of the noise response in the following calculation: yNLterm  Ze x2 ¼ Ze ðX1 cos x1 t þ X2 cos x2 tÞ2

¼ Ze ðX12 cos2 x1 t þ X22 cos2 x2 t þ 2X1 X2 cos x1 t: cos x2 tÞ  y1 þ y2 þ y3

ð3:72Þ

Similarly, the first two terms y1 and y2 provide two additional second-order harmonics of the frequencies 2x1 and 2x2, respectively. Furthermore, one obtains the third term y3 that generates sidebands of x1 and x2 after a few trigonometric calculating steps. y3 ¼ Ze X1 X2 ð2 cos x1 t: cos x2 tÞ

¼ Ze X1 X2 ½cosðx1 t  x2 tÞ þ cosðx1 t þ x2 tÞ

¼ Ze X1 X2 ½cosðx1  x2 Þt þ cos ðx1 þ x2 Þt

 Ze X1 X2 ðcos xLSB t þ cos xUSB tÞ

ð3:73Þ

with xUSB ¼ x1 þ x2 ; xLSB ¼ x1  x2

ð3:74Þ

where xUSB is the upper noise sideband frequency (rad/s); xLSB is the lower noise sideband frequency (rad/s). Thus, 1 2

x1 ¼ ðxUSB þ xLSB Þ  xc ¼ 2pfc ;

ð3:75Þ

with fc is defined by the carrier frequency (1/s); 1 2

x2 ¼ ðxUSB  xLSB Þ  xm ¼ 2pfm ; with fm is defined by the modulation frequency (1/s).

ð3:76Þ

3.4 Frequency Modulations of Nonlinear Aeroacoustics

39

carrier frequency modulation frequency

Fig. 3.9 Double sidebands of noise frequency modulations

1

Am

0.5

modulation frequency f m

0 -0.5 -1 0

1

2

3

4

5

6

7

8

9 -3

x 10

2

Ac

1 0

carrier frequency f c

-1 -2 0

1

2

3

4

fc

4

5

6

7

8

9 -3

x 10

2fm

A sb

2

sidebands

0 -2 -4 0

1

2

3

4

5

6

7

t(s)

8

9 -3

x 10

Fig. 3.10 Double sideband (DSB) of the amplitude modulation

Figure 3.9 shows the noise sideband frequencies of the modulation of the acoustic propagation waves. Due to the acoustic nonlinearity, the lower and upper sidebands are induced by the amplitude and frequency modulations of fc and fm. In fact, the noise sideband frequency varies in a bandwidth of frequency Dfm, e.g., in case of oil whirl. Therefore, the noise sideband frequencies fLSB and fUSB scatter in the corresponding frequency bandwidth Dfm. In the radio engineering, the carrier frequency fc is much higher than the modulation frequency fm (fc  fm). To broadcast the modulation signal in a long distance, the carrier signal modulates it in the double sideband (DSB) containing the lower (LSB) and upper sidebands (USB) with the ultra-high frequencies fLSB and fUSB (see Figs. 3.9 and 3.10). The noise sideband frequencies are modulated by the carrier and modulation frequencies in the wideband of frequency.

40

3 Acoustic Propagation Theory Tc =

1 fc

1 1 Tsb = Tm = 2 2 fm

4 3 2 Asb

1 0 -1 -2 -3 -4 0

1

2

3

4

5

t(s)

6

7

8

9

x 10

-3

Fig. 3.11 Frequencies and periods of the sideband signal

The modulation frequency fm is normally the human voice frequency spreading from 300 Hz to 3 kHz; therefore, the sidebands are generated with the maximum frequency width of the DSB of 2 fm about 6 kHz. The modulation signal is recovered at receiving by using demodulation or detection of the received sideband signals where the signal detector and low-pass filter are used. The entire process of sending and receiving signal is carried out by the MODEM method that consists of Modulation and DEModulation of the signals, generally applied in the wideband technique. The sideband signal generated by the carrier and modulating signals is displayed in Fig. 3.9, in which one finds the carrier and modulation frequencies again in the sideband signal. The envelop signal of the sideband given in Eq. (3.73) has a frequency of 2 fm since its period equals one-half of the period of the modulation signal, as shown in Fig. 3.11. The lower and upper noise sideband frequencies of fLSB and fUSB are calculated from the carrier fc and modulation frequencies fm with fm  fc; hence, f1  f2: • Lower noise sideband frequency (LSB):

fLSB ¼ fc  fm  f1  f2

ð3:77Þ

• Upper noise sideband frequency (USB):

fUSB ¼ fc þ fm  f1 þ f2

ð3:78Þ

To sum up, the nonlinear acoustic system induces some new additional terms due to the amplitude and frequency modulations as follows:

41

Misalignment

Unbalance whistle

Frequency order

3.4 Frequency Modulations of Nonlinear Aeroacoustics

Constant tone:

Fig. 3.12 Noise sideband frequencies in Waterfall plot (courtesy BMTS)

• The rectification term 12Ze X 2 in Eq. (3.70); • Second-order harmonic frequencies of 2x1 and 2x2 with the amplitudes X12 and X22 in Eqs. (3.70) and (3.72); • Noise sideband frequencies modulated by the excitation frequencies of x1 and x2 have the amplitudes that are proportional to the term of Ze*X1X2. According to Eqs. (3.73 and 3.74), the noise sideband frequencies are the sum or difference of the frequency components; hence, the first-order noise sideband frequencies result in 3x2 - 2x1, 2x2 - x1, 2x1 - x2, 3x1 - 2x2, 4x1 - 3x2, etc., (see Eqs. 3.80, 3.81), and Fig. 3.15. In the case of the unbalanced and misaligned rotor, the sidebands induced by the frequency modulations of misalignment, unbalance whistle, and constant tone (inner oil whirl) are indicated in the noise spectra Waterfall plot, as displayed in Fig. 3.12. Beat phenomenon, a special case of the modulation, is caused by the modulation of two harmonic vibrations with the frequency components of x1 and x2 that are close to each other. If their amplitudes are equal, only the amplitude modulation occurs. Otherwise, both amplitude and frequency modulations take place at the same time. The input signal is the sum of two sinus functions with the amplitudes a1 = a2 : a. x ¼ a1 sin x1 t þ a2 sin x2 t

¼ aðsin x1 t þ sin x2 tÞ h i h i 1 1 ¼ 2a sin ðx1 þ x2 Þ t  cos ðx1  x2 Þ t 2

2

 2a sin xc t  cos xm t

where xc ¼ 12ðx1 þ x2 Þ ¼ 2pfc ; fc xm ¼ 12ðx1  x2 Þ ¼ 2pfm ; fm

is the carrier frequency. is the modulation frequency.

ð3:79Þ

42

3 Acoustic Propagation Theory

A

f m ≡ 12 ( f1 − f 2 )

92.5 Hz

f c = 12 ( f1 + f 2 ) 85 Hz

1

100 Hz

fm

7.5 Hz

fm

7.5 Hz 7.5 Hz

0

fm

f2

fc

f1

f =

ε 2π

Fig. 3.13 Beat frequency 2 fm of the frequencies f1 and f2

Tc =

1 = fc

1 1 2

( f 2 + f1 )

Tb =

= 0.011s

1 1 = = 0.067 s 2 f m ( f1 − f 2 )

2 1.5 1

Asb

0.5 0 -0.5 -1 -1.5 -2 0

0.05

0.1

0.15

0.2 t(s)

0.25

0.3

0.35

0.4

Fig. 3.14 Beat response in the time domain

To demonstrate the beat frequency, one chooses two frequency components of f1 = 100 Hz and f2 = 85 Hz for the beat modulation that is shown in the frequency domain (Fig. 3.13). The carrier frequency fc is the average frequencies of f1 and f2 and equals 92.5 Hz, and the modulation frequency fm equals 7.5 Hz. The beat frequency of 2 fm (15 Hz) is defined by the frequency difference of f1 and f2. Figure 3.14 indicates that the closer the frequency components, the smaller the beat frequency; obviously, the larger the period of the beat signal. The beat period is inversely related to the beat frequency 2 fm; hence, the beat period is about 0.067 s at the beat frequency of 15 Hz. The carrier frequency has a period of 0.011 s at 92.5 Hz in the time domain. The generated first-order frequencies of the LSB and USB sidebands are resulted from the sum or difference of the frequency components of f1 and f2, which are located on the left- and right-hand sides of the carrier frequency fc, as plotted in Fig. 3.15.

3.4 Frequency Modulations of Nonlinear Aeroacoustics A

fc

LSB

43

USB

2 f m ≡ f1 − f 2

f1

fc =

f2

fm

1 2

f1 + f 2

fm fm

0

fm

( f1 − 4 f m )

f =

( f1 + 4 f m )

f1

ω 2π

( f1 − 6 f m ) ( f1 − 2 f m ) ( f1 + 2 f m ) ( f1 + 6 f m )

Fig. 3.15 LSB and USB noise sideband frequencies

Table 3.1 Noise sideband frequency modulations Lower sideband (LSB)

fc

2f2 - f 1

(f1 + f2)/2

3f2 - f 1 4f2 - f 1

f2 2f2

3f2

Upper sideband (USB)

2f1

(f1 + f2)

2f2 + f 1

fLSB

and

fUSB

3(f1 + f2)/2

2f1 - f 2

f1

2f1 + f 2

1X

3f1 - f 2 3f1

4f1 - f 2

8 f1  2fm ¼ fc  fm ¼ f2 > > < f1  4fm ¼ fc  3fm ¼ 2f2  f1 ¼ f  6fm ¼ fc  5fm ¼ 3f2  2f1 > > : 1 ... 8 f1 þ 0fm > > > > < f1 þ 2fm ¼ f1 þ 4fm > > f þ 6fm > > : 1 ...

f Order

¼ fc þ fm ¼ f1 ¼ fc þ 3fm ¼ 2f1  f2 ¼ fc þ 5fm ¼ 3f1  2f2 ¼ fc þ 7fm ¼ 4f1  3f2

2X 3X

ð3:80Þ

ð3:81Þ

However, the nonlinear terms of the noise response are not only proportional to the input amplitude squared x2 but also to the high-order input amplitude, such as x3 or higher orders compared to Eq. (3.64). According to Ehrich [9], the truncation of the modulation frequencies of f1 and f2 could produce two new second harmonic frequencies of (f1 - f2) and (f1 ? f2) that in turn generate the third harmonics of (2f2 ? f1) and (2f1 ? f2) in a strongly nonlinear acoustic system, as demonstrated in Table 3.1. To generate the LSB frequencies, one superimposes the excitation frequency f2 on (2f2 - f1) to obtain the second-order frequency (3f2 - f1) following the thin arrow, and superimposing f2 again on the second-order frequency to generate the

44

3 Acoustic Propagation Theory

third-order frequency (4f2 - f1). Another way, following the dark arrow, one obtains the new high-order harmonic frequencies of 2f2 and 3f2 by adding f2 to itself and 2f2, respectively. In order to receive the third-order frequency (2f2 ? f1), one does the same way by adding f2 to (f1 ? f2) in the direction of the dashed arrow. Similarly, the USB frequencies are generated in the same way.

References 1. Fahy, F., Gardonio, P.: Sound and Structural Vibration, 2nd edn, Academic Press, London (2007) 2. Crocker, M. J.: Noise and Vibration Control, Wiley, NY (2007) 3. Dawkins, S.: A guide to aeroacoustics: An overview, Lighthill’s equation, related model equations, Webster’s Digital Services (2011) 4. Schobeiri, M.: Turbomachinery Flow Physics and Dynamic Performance, 2nd edn, Springer, Berlin (2012) 5. Nguyen-Schäfer, H.: Rotordynamics of Automotive Turbochargers. Springer, Berlin (2012) 6. Nguyen-Schäfer, H. Kleinschmidt, R.: Analysis and nonlinear rotordynamics computation of constant tone in automotive turbochargers. 17th ATK Conference, Dresden (2012) 7. Tam, C.: Computational Aeroacoustics: A Wave Number Approach, Cambridge University Press, Cambridge (2012) 8. LMS Virtual Lab Acoustics: http://www.lmsintl.com/acoustic-simulation (2012) 9. Ehrich, F.: Handbook of Rotordynamics, Krieger Publishing Company, Florida (2004) 10. Software MSC: Program ASTRAN of Aero and Vibroacoustics (2012) http://www.mscsoft ware.com/Products/CAE-Tools/Actran-For-Nastran.aspx

Chapter 4

Analyzing Root Causes of Noise

The induced noise types of automotive turbochargers in passenger vehicles have been already displayed in Chap. 2. Aerodynamic noise types, such as pulsation, rotational, growling, and whining noise, are generated from airflows in compressors of turbochargers by interactions between the fluid (charge air), solid surface (turbocharger), and fluid (ambient air) at their interfaces. Rotordynamic noise types like unbalance whistle, constant tone, high-order harmonic noise, and wear noise are created from rotordynamics of turbochargers by interactions between the fluid (oil), solid surface (turbocharger), and fluid (ambient air) at their interfaces. The unbalance whistle is generated by the rotor unbalance due to production tolerances; the constant tone is induced by the inner oil whirl occurring in the oil film bearings due to self-excitation instability. To improve the noise behavior of turbochargers in the vehicles, the root causes of these noise types should be analyzed in the following sections.

4.1 Aerodynamic Noise Pulsation whistle, rotation, growling, and whining noise are the typical aerodynamic noise types in automotive turbochargers.

4.1.1 Root Causes of Aerodynamic Noise Pulsation whistle is caused by slight differences in the compressor wheel (CW) chamber volumes due to milling or molding process variations. Due to the different chamber volumes, charge air pressures are slightly different at the blade outlets, as shown in Fig. 4.1. The generated charge air pressures in the CW chambers are substituted at the compressor outlet in every rotation cycle. This pressure perturbation dp causes the synchronous pulsation whistle that is H. Nguyen-Schäfer, Aero and Vibroacoustics of Automotive Turbochargers, DOI: 10.1007/978-3-642-35070-2_4, Ó Springer-Verlag Berlin Heidelberg 2013

45

46 Fig. 4.1 Pressure distributions at the compressor wheel (CW) outlet

4 Analyzing Root Causes of Noise

ω

CW chambers

V*>V

V

pV * = pV + δ p pV

propagated through the compressor housing, charge air intercooler, and charge air duct. As a result, the charge air system is excited by the synchronous pulsation whistle, and radiates the pulsation whistle as airborne noise into the car cabin and environment. To reduce the pulsation whistle, a pulsation damper based on the Helmholtz’s resonator is installed behind the compressor outlet in the charge air duct. Thus, the pulsation whistle amplitude is reduced in the damper. Rotational noise is generated by rotating blades of the compressor or turbine wheels. In the case of a radial compressor, the charge air pressure is built in the compressor wheel chambers due to rotation. The rotating blades disturb the charge air pressure zone at the blade inlet, blade outlet, and near the compressor tongue in each rotation cycle. This causes air wakes and vortexes behind the rotating blades and near the compressor tongue, leading to local pressure disturbances that induce rotational noise in the turbocharger. The frequency of the rotational noise fRN equals the product of the CW blades Z and rotor frequency fR = x/(2p), also fRN = fR.Z. At the compressor inlet, Z equals the number of the main blades (Z = 6); at the compressor outlet, Z equals the number of the main and splitter blades (Z = 12), as displayed in Fig. 4.1. The Waterfall plot measured at the compressor housing indicates that the rotational noise has sixth frequency order (6X) at the compressor inlet and 12th frequency order (12X) at the compressor outlet, as shown in Fig. 4.2. Generally, either a large number of the CW blades or an optimized geometry of the compressor tongue is used to reduce the rotational noise. The first one increases the noise frequencies (6 and 12X) above 16 kHz so that adult human being ears cannot recognize them, but animals could hear such noises at high frequencies. The optimized tongue at the compressor outlet reduces pressure disturbances (e.g., air wakes and vortexes) at the tongue. Therefore, the charge air pressure smoothly flows into the compressor outlet in each rotation cycle, causing little rotational noise. Growling noise (compressor stall-related noise) is induced by the partial reversal of the charge air at the suction side near the blade outlet. Whining noise (compressor surge-related noise) is produced by the deep surge condition in the compressor where the charge air totally recirculates from the

4.1 Aerodynamic Noise

47

Fig. 4.2 Waterfall plot of noise spectra in turbochargers

compressor outlet to compressor inlet. The whining noise occurs when car driver suddenly releases gas pedal to decelerate the vehicle. At the moment, the currently required charge air mass flow rate for the engine, still at high turbocharger speed, is strongly reduced because the engine throttle valve begins to close. It leads to the pressure increase at the compressor outlet due to the still high turbocharger speed. This causes the deep surge working condition in the compressor, leading to whining noise in the turbocharger. To better comprehend the root causes of the compressor rotating stall and surge, flow behaviors in the compressor must be studied and analyzed in the following subsections.

4.1.2 Aerodynamics of Radial Compressors Figure 4.3 shows a backward swept radial compressor wheel that is mostly used in automotive turbochargers in order to improve the compressor performance map width. Note that the rotating direction x of the CW is against the blade outlet direction; i.e., the outflow direction of the radial CW is against the circumferential velocity at the blade outlet. The radial CW consists of main and splitter blades that divide the CW chambers into two equal chambers. The splitter blade has two functions: first, avoiding choke flow at the blade inlet 1; second, reducing flow separation at the suction side near the blade outlet 2. However, splitter blades cause more flow friction in the CW and possibly decreasing mass flow rate in the compressor.

48

4 Analyzing Root Causes of Noise flow chamber

ω >0

shroud

ω >0

2

main blade

2 flow direction (1 → 2)

hub

1

1

splitter blade

Fig. 4.3 Backward swept radial compressor wheel (CW)

Shroud is the contour on the top of the blades where the air gap is between the shroud and compressor housing. Over the shroud the airflow could leak from the pressure side to the suction side of the blades during rotating. Hub is the bottom surface (conical shape) located between the main and splitter blades from the blade inlet 1 to the blade outlet 2, as indicated in Fig. 4.3. Three-dimensional flows in the CW are limited in the flow pattern between the main and splitter blades, hub, shroud, and as well as compressor housing. The compressor setup of a typical automotive turbocharger is displayed in Fig. 4.4. The compressor contains the CW with the super back, diffuser, volute, and compressor housing. Intake air enters the blade inlet 1 and is compressed along the blades from the compressor inlet 1 to the compressor outlet 2, and from the hub to shroud of the impeller (blade). Due to high rotational speed of the CW, the intake air pressure is increased at the blade outlet 2. The charge air pressure is Fig. 4.4 Radial compressor setup

volute

diffuser

2 hub

shroud

back face (super back)

compressor housing impeller

1 flow direction

4.1 Aerodynamic Noise

49

additionally gained in the diffuser built in the compressor housing between the CW and the volute. The charge air mass flow rate through the compressor, not the charge air pressure, is used for turbocharging downsized engines. The compressor torque is derived from the angular momentum equation. M C ¼ m_ C ðr2 cu;2  r1 cu;1 Þ

ð4:1aÞ

where m_ C is the compressor mass flow rate; r1 and r2 are the inlet (inducer) and outlet (exducer) radii of CW; cu,1 and cu,2 are the tangential (or whirl) velocities at the inlet and outlet. The compressor mass flow rate is resulted from the air density q, meridional velocity component cm, and compressor cross-sectional area AC. m_ C ¼ qcm AC

ð4:1bÞ

The compressor power is resulted from the compressor torque and angular rotor speed. PC ¼ M C x ¼ m_ C x ðr2 cu;2  r1 cu;1 Þ ¼ m_ C ðu2 cu;2  u1 cu;1 Þ

ð4:2aÞ

within u1 and u2 are the circumferential velocities at the CW inlet and outlet. The compressor power is calculated from the turbine power PT and mechanical efficiency gm. " #  ðj1 p4 j Þ g PC ¼ gm PT ¼ gm gT m_ T cp;g T3 1  ð4:2bÞ p3 where gT is is T3 is p3 is p4 is jg cp,g is

the the the the the the

turbine efficiency; exhaust gas inlet temperature; exhaust gas inlet absolute pressure; exhaust gas outlet absolute pressure; exhaust gas isentropic coefficient (jg & 1.32); exhaust gas heat capacity at constant pressure.

The required compressor power for the charge air compression is written in a function of the mass flow rate, inlet temperature, and pressure ratio of the compressor. "  j1 # PC;ideal m_ C cp;a T1 p2 ð j Þ a PC ¼  1 ð4:2cÞ gC gC p1

50

4 Analyzing Root Causes of Noise

where is gC is T1 is p1 p2 is is ja cp,a is

the the the the the the

compressor efficiency; air inlet temperature; air inlet absolute pressure; charge air outlet absolute pressure; charge air isentropic coefficient (ja & 1.4); charge air heat capacity at constant pressure.

Having combined Eqs. (4.2b) and (4.2c), one obtains the pressure ratio of the compressor pC in the first turbocharger equation. p2 pC  ¼ p1

j " #!ðj1 Þa  ðj1  j Þg cp;g p3 m_ T T3 1þ g 1 p4 m_ C T1 TC cp;a



ð4:2dÞ

Within the overall turbocharger efficiency is defined by gTC ¼ gm gT gC

ð4:2eÞ

Convention for the flow and blade angles of ai and bi: • ai [ 0 if cu,i [ 0; cu,i has the same direction of the circumferential velocity ui; • bi \0 if wu,i \ 0; wu,i is opposite to the circumferential velocity ui. The absolute gas velocity in the CW is the sum of the relative and circumferential gas velocities wi and ui, respectively, as shown in Fig. 4.5 ~ ci ¼ ~ wi þ ~ ui

ð4:3Þ

Therefore, the relative velocities are calculated for the CW inlet and outlet for i = 1, 2. w2i ¼ c2i þ u2i  2~ ci~ ui ¼ c2i þ u2i  2ci ui cosð~ ci ;~ ui Þ p 2

¼ c2i þ u2i  2ci ui cosð  ai Þ ¼ c2i þ u2i  2ui ci sin ai

ð4:4Þ

The tangential vector component cu,i for i = 1; 2 are calculated from the velocity triangle map, as given in Fig. 4.5. cu;i ¼ ci sin ai

ð4:5Þ

Thus, the relative velocity components wi for i = 1; 2 are rewritten by substituting Eq. (4.5) into Eq. (4.4) in qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4:6Þ jwi j ¼ c2i þ u2i  2ui cu;i

Having combined Eqs. (4.2a) and (4.6), one obtains the compressor specific work WC in the Euler’s turbocharger equation. This specific work WC is defined by the compressor power per unit mass flow rate in the unit of W/(kg.s-1) or J/kg.

4.1 Aerodynamic Noise

51

ω >0

→

β1 0

2

β2

→

cu , 2 > 0

wu , 2 < 0 cu ,1 > 0 wu ,1 < 0

0

1

r2

→

ci = wi + ui

r1

→

→ 1

u1 β1 < 0

c

→

α1 > 0

u2 → 2

→

w1 →

cm , 2

c

α2 > 0

w2 β2 < 0

Fig. 4.5 Velocity triangle map of a radial compressor wheel (CW)

The Euler’s turbocharger equation is formulated in PC ¼ ðu2 cu;2  u1 cu;1 Þ m_ C  1 ¼ ðc22  c21 Þ þ ðu22  u21 Þ  ðw22  w21 Þ 2

WC 

where

ð4:7Þ

w2 \ w1 because the outlet air density is larger than the inlet air density in case of compressors. Note that to avoid the flow separation at the suction side on the compressor blades, the velocity ratio (w2/w1) should lie between 0.65 and 0.75 if the compressor performance is fulfilled according to De Haller’s rule [1, 8]. The tangential component cu,1 of the inlet absolute velocity is computed from the inlet absolute velocity c1, and its inlet flow angle a1 (a1 [ 0), as given in Fig. 4.5. cu;1 ¼ c1 sin a1

ð4:8Þ

The tangential component cu,2 of the outlet absolute velocity is resulted from the outlet circumferential velocity u2, meridional velocity component cm,2, and outlet blade angle b2 (b2 \ 0), as shown in Fig. 4.5. cu;2 ¼ c2 sin a2

ð4:9Þ

¼ u2 þ cm;2 tan b2

Having applied the second law of thermodynamics for a steady-state adiabatic process, one obtains the compressor specific work WC ¼ u2 cu;2  u1 cu;1 ¼ ht;2  ht;1

¼

1 cp ðT2  T1 Þ þ ðc22 2



c21 Þ

ð4:10Þ

52

4 Analyzing Root Causes of Noise

(a)

(b)

ω>0

separation zone

suction side

2

2

→

w

1 1

jet

+ + + pressure side

wake

(c)

backward flow forward flow near shroud near hub 2 →

1

w

forward flow

Fig. 4.6 Flow velocity profiles in a radial compressor wheel

where ht,1, ht,2 cp c

total enthalpies at the CW inlet and outlet (see Appendix A); air heat capacity at constant pressure; absolute air velocity.

In the case of the backward swept CW mostly used in automotive turbochargers, the charge air is compressed on the impeller front face in the rotating direction x, which is called the pressure side (ps = ? + +), as indicated in Fig. 4.6a. On the contrary, the airflow separates from the impeller back face near the outlet 2, which is called the suction side (ss = - - -) opposite to the pressure side. On the pressure side, the airflow jet occurs at the blade outlet 2 and remains on the blade with the forward flow during the rotation at a velocity x (see Fig. 4.6b). On the suction side opposite to the pressure side, the charge air pressure decreases from the blade inlet 1 to the blade outlet 2. Therefore, the air wake begins separating from the suction side near the shroud where the relative velocity gradient at the wall equals zero. In the separation zone, the charge air partially recirculates at the negative velocity gradient at the wall; however, the volumetric flow rate in the CW is still positive with the forward flow. The air outflow is mostly forward even at the suction side near the hub. Note that the flow profile at the flow pattern outlet changes from the hub to shroud, as shown in Fig. 4.6c. When the charge air partially separates from the suction side near the blade outlet, in the vaneless diffuser, or near the inducer tips of the CW, the rotating stall (also mild surge) happens (see Figs. 4.6 and 4.7). In this case, the mass flow rate is reduced but still positive in the compressor, causing however the growling noise.

4.1 Aerodynamic Noise

53

ω >0 (a)

suction side (ss)

(b) 2

flow separation at impeller inducer

ω >0

(ss) 1

2

+ + + pressure side (ps)

→

w1

1

→ 2

w

Fig. 4.7 Flow reversal in the impeller inducer

In contrast to the rotating stall, the charge air totally recirculates to the compressor outlet to the compressor inlet, from the suction side to the pressure side, and from the compressor hub to the compressor shroud at low air mass flow rates and high pressures, as shown in Fig. 4.8. This phenomenon is called the compressor deep surge that mostly follows the rotating stall in case the pressure at the compressor outlet is much higher than the charge air pressure. In the deep surge, the mass flow rate takes turns periodically changing from the positive (forward flow) to negative (backward flow) mass flow rates. During the periodic airflow reversal (recirculation), very loud whining noise takes place in the compressor. Due to friction in the airflow recirculation, air temperature is strongly increased in the CW. Moreover, the deep surge causes damage the thrust bearing and turbocharger as well. Note that the compressor deep surge could occur in the diffuser when the pressure at the compressor outlet is relatively high at low mass flow rates so that the flow recirculates from the diffuser to the compressor inlet. In the deep surge, the compressor does not transport any

. mC = 0 suction side

deep surge (total flow reversal)

pressure side

Fig. 4.8 Flow reversal in a radial compressor wheel

54

4 Analyzing Root Causes of Noise

π C ,tt =

p2 t p1t surge line

3

full load curve DP

M1

75%

Pnom Nmax

70%

2

67% 65% N2

ηC=60%

1

choke line NTC

N1

0

.

.

.

mchoke mC ,cor

m T = C 1t p1t

Fig. 4.9 Compressor performance map

mass flow rate at all because the airflow entirely recirculates from the compressor outlet to the compressor inlet and vice versa, as shown in Fig. 4.8. These phenomena of the compressor rotating stall and deep surge will be analytically analyzed in the following subsection.

4.1.3 Stall and Surge in Radial Compressors Flow instabilities in the compressor are to blame for rotating stall and surge mostly in the working condition at small mass flow rates and high pressures. The flow instabilities cause flow reversals that happen at certain operating points left from the surge line in the compressor performance map. The compressor ratio pC versus corrected mass flow rate is displayed in the performance map (see Fig. 4.9). At low rotor speeds from the idle condition (about 30 % of Nmax), the compressor pressure ratio increases with a relatively high pressure gradient in the full load curve to ensure a good transient response in low-end torque (LET). At further increasing the rotor speed N to nearly 70 % of Nmax, the compressor pressure ratio increases to approximately 2.5, and the engine achieves the maximum torque M1. The nominal power Pnom occurs just before the maximum operating rotor speed Nmax. The maximum rotor speed corresponding with u2,max = 560 m/s is reached in highland applications where the rotor speed increases by 8–10 % (highland speed reserve) above the nominal operating rotor speed. The compressor pressure ratio is formulated in a function of the corrected mass flow rate and turbocharger speed in the performance map.

4.1 Aerodynamic Noise

55

pC  where pC p1, p 2 m_ C

p2 ¼ f ðm_ C ; NTC Þ p1

ð4:11Þ

compressor pressure ratio; air pressures at the compressor inlet and outlet; mass flow rates in the compressor.

At decreasing compressor volumetric flow rates, the charge air pressure rises in the turbocharger isospeed NTC, as shown in Fig. 4.10. As the turbocharger isospeed passes the surge line, the compressor rotating stall occurs where the charge air partly recirculates in the CW periodically. In the stall condition, the compressor pressure ratio decreases with reducing volumetric flow rates; however, the volumetric flow rates in the compressor are still positive. The charge air partly recirculates from the compressor outlet to the compressor inlet only in a short time and is pumped again to the compressor outlet; thus, the rotating stall period is relatively short. In contrast, in the deep surge condition the compressor volumetric flow rate decreases in the direction of negative volumetric flow rates at further reducing compressor pressure ratios. Suddenly, the duct pressure behind the compressor drops below the compressor inlet pressure; thus, the charge air is compressed again to the compressor outlet in a normal condition on the rhs of the surge line (see Fig. 4.10). In this case, the surge period lasts much longer than the rotating stall period because the large duct volume behind the compressor must be completely discharged to a minimum pressure that is below the current compressor inlet pressure. Then, the duct is charged again with the compressed charge air; its pressure increases again above the entering charge air pressure, leading to compressor stall and surge. As a result, the surge process is periodically repeated in the . VC < 0

deep surge →

w

. VC > 0

πC

mild surge / rotating stall

ω

→

w

1 0

Fig. 4.10 Compressor stall and surge in the performance map

.

VC

56

4 Analyzing Root Causes of Noise

surge working condition. Both rotating stall and surge flow conditions cause the very disturbing growling and whining noises, respectively for driver and passengers in the vehicle cabin. Additionally, the charge air temperature is drastically increased in the compressor due to repeating periodic flow reversals. In this case, the thrust load acting on the thrust bearing takes turns changing from the turbine wheel to CW direction and vice versa due to the very large pressure pulsations of the charge air, leading to damage of the thrust bearing and the whole turbocharger as well. Note that compressor surge has been considered in one-dimensional flow pattern; however, it could be turned into the three-dimensional rotating stall instability, as shown in Fig. 4.8. At first, mild surge occurs when the operating flow condition in the turbocharger isospeed passing the surge line at reducing volumetric flow rates. There begins the airflow to separate from the suction side near the blade outlet. Usually, the deep surge follows the mild surge (rotating stall) directly when the entire flow reversal takes place in the compressor. However, deep surge and rotating stall could take turns occurring in certain operating conditions. To improve the flow stabilities for the stall and surge behaviors in radial compressors, the outlet blade angle |b2| should be increased (i.e., more backward swept blades). However, the larger the backward swept blade angle is, the higher the flow resistance exists in the CW, causing the compressor efficiency reduction. Additionally, the normal stresses caused by the centrifugal force acting on the blades also increase due to the longer blade length, leading to blade damage in some cases. In the following section, the thresholds of instability of the compressor rotating stall and surge are analyzed and discussed. At first, some useful dimensionless numbers and coefficients are defined for analyzing the compressor stall and surge instabilities in the performance map. The flow coefficient / is defined by the ratio of the meridional velocity to the circumferential velocity at the blade outlet 2. /

cm;2 V_ C m_ C / 2 / u2 D2 u2 q2 D22 u2

ð4:12Þ

The pressure rise coefficient w is defined by the ratio of the pressure increase in the compressor to the kinetic energy at the blade outlet 2. w

DpC DpC pC / / 2 2 2 q2 NTC D2 q2 u2 q2 u22

ð4:13Þ

The compressor characteristic C is the behavior between the pressure rise and flow coefficients, as shown in Figs. 4.10 and 4.13. The throttle characteristic T of the compressor system containing the duct, plenum, and throttle (see Fig. 4.12) describes the volumetric flow rate in the compressor depending on the throttle coefficient aT and the pressure drop DpT.

4.1 Aerodynamic Noise

Ψ=

∆ pC ρ 2u22

57

static instability if

static stability if

C '> T '

C '< T '

(T) M

throttle characteristic (T)

A

(C) (C)

B (T) compressor characteristic (C)

.

φ=

VC D22u2

Fig. 4.11 Static instability operating condition

V_ T / aT AT

pffiffiffiffiffiffiffiffi DpT

ð4:14aÞ

The relation between the flow and pressure rise coefficients is resulted from Eqs. (4.12), (4.13), and (4.14a). wT /

1 2

ðAT aT Þ

/2T

ð4:14bÞ

Figure 4.11 shows that the stable condition (without stall and surge) locates on the rhs of the surge line. Left from the surge line, the compressor characteristic is statically stable between M and A in the isospeed NTC. Then, it becomes statically unstable between A and B when the compressor characteristic slope C0 is larger than the throttle characteristic slope T0 according to [3].     ow ow 0 0 C [T  ð4:15Þ o/ C o/ T By differentiating Eq. (4.14b), one obtains the throttle characteristic slope T0 that is always positive.   ow 2/T 2n2 / T0  ¼ T2 T [ 0 / ð4:16Þ 2 o/ T ðAT aT Þ AT where aT is the throttle coefficient; nT is the pressure loss coefficient of the throttle (nT : 1/aT).

58

4 Analyzing Root Causes of Noise compressor (C)

duct (LD)

plenum (VP)

p2 cP pD

AP

Helmholtz resonator

AT throttle (T)

p1 Fig. 4.12 Compressor system setup

Equation (4.16) indicates that the throttle characteristic slope T0 is proportional to the throttle flow coefficient /T . Therefore, the lower the throttle flow coefficient /T , the smaller the throttle characteristic slope T0 is. The volumetric flow rate in the throttle is limited by the throttle area AT (see Fig. 4.12). If the compressor volumetric flow rate ð/C  C 0 Þ is larger than the maximum allowed volumetric flow rate in the throttle ð/T  T 0 Þ, the airflow reverses in the compressor. In case of C0 [ T0 , as given in Eq. (4.15), the static flow instability takes place in the compressor. In this case, the airflow separates from the suction side near the blade outlet and recirculates to the compressor inlet, leading to compressor rotating stall and deep surge. Logically, the compressor working condition is statically stable only if C0 \T 0

ð4:17Þ

In this case, the compressor volumetric flow rate is less than the maximum allowed volumetric flow rate in the throttle; thus, the compressor operates in a statically stable condition and satisfies the stability condition given in Eq. (4.17). Both statically unstable and stable conditions are displayed in the compressor performance map, as shown in Fig. 4.11. The threshold of static instability (onset of static instability) takes place if the compressor characteristic slope C0 equals the throttle characteristic slope T0 . In this working condition, the airflow begins separating from the suction side near the blade outlet. C0 ¼ T 0

ð4:18Þ

At the static instability, as given in Eq. (4.15), the airflow collapses and separates from the blade. The limited small volumetric flow rate in the throttle is to blame for the static instability. On the contrary, the negative damping of the airflow vibration in the compressor system causes dynamic instability [1]. In the

4.1 Aerodynamic Noise

59

following section, the dynamic instability behavior of the compressor is thoroughly analyzed and discussed. Figure 4.12 shows the compressor system containing the compressor C, duct with a length LD and cross-sectional area AP, plenum with a volume VP and sound velocity cP, and throttle T. Due to pressure reflection at the throttle, airflow vibrates in the compressor system with amplitude f(t) and angular frequency x. The homogenous vibration equation of the characteristics of the mass flow rate, perturbed pressure, and density is written for the compressor system. _ þ S fðtÞ ¼ 0 M €fðtÞ þ D fðtÞ

ð4:19Þ

Within f(t) is a perturbation characteristic, M, D, and S are the vibration equation coefficients computed from the flow rate equations of the compressor system [1]. The system-mass coefficient M is defined by   L D VP M T0 [ 0 AP c2P The system-damping coefficient D is written in   LD C 0 T 0 VP D  AP c2P

ð4:20Þ

ð4:21Þ

The system-stiffness coefficient S is given by S  ðT 0  C 0 Þ

ð4:22Þ

Equation (4.19) is a linear second-order ordinary differential equation (ODE) whose homogeneous solution results in fðtÞ ¼ Aekt

¼ AeðaþjxÞt ¼ Aea t ejxt at

¼ Ae ðcos x t þ j sin x tÞ

where A is k is x is j is

the the the the

ð4:23Þ

vibration response amplitude of f(t); complex eigenvalue of the vibration system (k = a ? jx); surge angular frequency; imaginary unit (j2 = -1).

Note that the system response f(t) is conditionally stable if the real term a of the complex eigenvalue k is negative; otherwise, at a [ 0 the solution exponentially increases with time. This leads to the system instability that causes rotating stall

60

4 Analyzing Root Causes of Noise

and surge. In the following section, the real term a (growth/decay rate) is computed. Having substituted the first and second derivatives of f(t) into Eq. (4.19), one obtains the characteristic equation of the vibration system [4]. Mk2 þ Dk þ S ¼ 0 The eigenvalue k is resulted from solving Eq. (4.24). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  ffi D D S  k ¼ a þ j:0 ¼   2M 2M M

ð4:24Þ

ð4:25Þ

The system-stiffness coefficient S in Eq. (4.25) must be positive so that the growth/decay rate a is always negative. Thus, S [ 0 satisfies the condition of a \ 0 that is necessary for the compressor static stability of C0 \ T0 . However, Eq. (4.25) can be formulated in the complex number when the term in the square root is negative. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2ffi D S D  j ð4:26Þ k ¼ a  jx ¼   2M M 2M Obviously, one obtains D a ¼  ;x ¼ 2M

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2ffi S D  M 2M

ð4:27Þ

To ensure that a \ 0, the system-damping coefficient D must be positive (see Appendix D); therefore, D [ 0 is required for the compressor dynamic stability condition, as given in Eq. (4.29). Figure 4.13 shows the stability behaviors of vibration responses at different values of a. The rotor response is stable at a \ 0 where the compressor surge amplitude exponentially dies down with time. The threshold of instability takes place at a = 0, where the vibration response amplitude is constant over time. Finally, the vibration response is unstable (compressor surge condition) at a [ 0, where the surge amplitude exponentially increases with time. Alternatively, the Routh-Hurwitz stability criterion can be used to analyze the flow stability of the compressor system vibration (see Appendix E) [4]. The airflow behavior is conditionally stable if all coefficients M, D, and S in Eq. (4.24) are nonzero and have the same positive or negative signs to satisfy the necessary conditions for stability. First, the first coefficient M is positive because the throttle characteristic slope T0 is always positive. Second, the positive system-stiffness coefficient S [ 0 fulfills the static stability condition of C0 \ T0 at a \ 0. S ¼ ðT 0  C 0 Þ [ 0 ) C0 \ T 0

ð4:28Þ

4.1 Aerodynamic Noise

61

stable 1

unstable

threshold of instability

(a): α < 0

(b): α = 0

1

(c): α > 0

60 40

0.5

0.5

0

0

-0.5

-0.5

x

20 0 -20 -40 -1

0

1

2

3

4

-1

0

1

2

3

4

-60

0

1

2

3

4

0

1

2

3

4

t(s) 1

1

0.5

0.5

0

0

-0.5

-0.5

60 40

y

20 0 -20 -40 -1

0

1

2

3

4

-1

0

1

2

3

4

-60

t(s)

r(t)

r(t)

0

0

0

40 20

0.5

y

0.5

-20

r(t)

-0.5 -0.5

-40 -0.5

0

0.5

1

-1

-0.5

0

0.5

1

-50

0

50

x

Fig. 4.13 Stability behaviors of vibration responses

The result of Eq. (4.28) indicates that the static stability condition given in Eq. (4.17) has been reconfirmed using the Routh-Hurwitz criterion (q.e.d.). Third, the dynamic stability condition of a \ 0 is satisfied if the systemdamping coefficient D in Eq. (4.24) is positive.   LD C0 T 0 VP D¼  [0 AP c2P ð4:29Þ LP c2P 1  ) C0 \ AP VP T 0 B02 T 0 where the factor B0 is defined by   AP VP 1 AP 2 ; ¼ B  LP c2P x2H LD 02

B0 in unit ½m:s:

The Helmholtz frequency of the plenum/duct is defined by rffiffiffiffiffiffiffiffiffiffiffi   AP x H  cP ; xH in unit s1 L D VP

ð4:30Þ

ð4:31Þ

The factor B0 changes inversely related to the Helmholtz frequency squared. The Helmholtz frequency becomes small in case of the large plenum volume VP, the long duct length LD, and the small cross-sectional area AP according to Eq. (4.31). Therefore, the factor B0 is quite large at the small Helmholtz frequency. At the large factor B0 , the stability condition in Eq. (4.29) is not fulfilled any longer.

62

4 Analyzing Root Causes of Noise

Therefore, the dynamic instability occurs in the compressor, leading to compressor surge. The undamped eigenfrequency (natural frequency) of the compressor system is computed from Eq. (4.19). rffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S AP T0 C0 xn  : ¼ cP M T0 LD VP ð4:32Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffi C0 ) xn ¼ xH 1  0 \ xH T This result indicates that the Helmholtz frequency xH of the compressor system is normally higher than the surge frequency x given in Eq. (4.27). Therefore, the compressor working condition becomes dynamically unstable at a [ 0 in a negative damping system according to Eq. (4.29). C0 [

1

ð4:33Þ

B02 T 0

The dynamic instability condition is much critical at large values B0 and T0 that lead to a very small rhs term in Eq. (4.33). In this case, the compressor characteristic slope C0 easily exceeds this rhs term although the compressor characteristic slope C0 is smaller than the throttle characteristic slope T0 , as shown in Fig. 4.14. As a result, the dynamic stability condition in Eq. (4.29) must be taken into account at first. Then, the static stability condition in Eq. (4.28) is considered. Generally, the factor B0 in unit [m s] is transformed into the dimensionless factor B [-] which was defined by Greitzer [3].

Ψ=

∆pC

dynamic instability if

ρ 2u22

C '>

1 B '2 T '

(T) M

A (C) throttle characteristic (T) compressor characteristic (C)

. VC φ= 2 D2 u2 Fig. 4.14 Dynamic instability operating condition

4.1 Aerodynamic Noise

63

 u2 u2 ¼ 2AP 2xH LD rffiffiffiffiffiffiffiffiffiffiffi u2 VP ¼ 2cP AP LD

B  B0 :



ð4:34Þ

within u2 is the circumferential velocity at the impeller outlet; cP is the sound velocity in the plenum. Experimental results have shown that the threshold of the dynamic instability (deep or mild surge) takes place in the compressor when the dimensionless factor B is between 0.7 and 0.8 [1].

4.2 Unbalance Whistle 4.2.1 Root Causes of Unbalance Whistle Production process of CW and turbine shaft causes an initial unbalance, in which the mass center does not exactly locate in the geometrical axis of the CW or turbine shaft, leading to unbalance forces acting on the rotor. Besides the initial unbalance due to producing tolerances, additional rotor unbalances are resulted from misaligned mounting the CW in the turbine shaft. This misalignment generates an unbalance moment acting upon the rotor [4]. Excessively large unbalance forces and moments induce high response amplitudes; these lead to bearing wear, seizure of the journal in the bearings, and rub contact between the compressor and turbine wheels and their housings. Moreover, the rotor unbalance generates the unbalance whistle whose frequency is synchronous with the rotor frequency (i.e., 1X synchronous noise). The unbalance force is proportional to the unbalance and rotor speed squared. This is a dipole noise force acting on the turbocharger and inducing the unbalance whistle. The unbalance whistle is one of the disturbing airborne noise types of automotive turbochargers. To reduce the unbalance whistle, the turbocharger rotor must be high-speed balanced at two balancing planes. There are two rotor balancing types of automotive turbochargers: • Low-speed balancing (shop balancing) is used for rigid rotors at a low balancing speed up to 3,400 rpm, depending on the balancing machine types, to reduce the initial unbalance caused by the production process. Generally, the whole rotor is not balanced at the shop balancing, but only the CW and turbine shaft are separately balanced at a rigid state with the low-speed balancing. Hence, it is also named single part balancing. The low-speed balancing is generally carried out in two balancing planes at the nose and back face of the wheel. The goal of

64

4 Analyzing Root Causes of Noise

the low-speed balancing is to reduce the unbalance excitations; therefore, this prevents bearing damage due to rub contact and seizure between the journal and radial bearings caused by an excessively unbalanced rotor. • High-speed balancing (trim balancing) is applied to flexible rotors at high balancing speeds of nearly 200,000 rpm, generally above the first bending critical speed. At high rotor speeds, the rotor deflects in the radial direction, causing an additional static unbalance. The static and mounting-related unbalances must be removed by the trim balancing in order to improve aeroacoustic qualities of automotive turbochargers for passenger vehicles. The high-speed balancing is carried out in two balancing planes at the CW screw-nut and hub surface between the blades near the CW outlet. The trim balancing purpose is to reduce the unbalance whistle noise level in passenger vehicles. Due to balancing costs, the trim balancing has been usually omitted in turbochargers for commercial vehicles and industrial applications unless the customers optionally request.

4.2.2 Unbalance Types of Turbochargers Manufacturing CW and turbine shaft (i.e., turbine wheel and rotor shaft) induces static, couple, and dynamic unbalances. The static unbalance occurs when the polar mass-inertia axis differs from the rotational axis by an eccentricity e. On the contrary, the couple unbalance occurs when the polar mass-inertia axis differs from the rotational axis by a misalignment angle a; however, they intersect each other at the mass center G. In the static unbalance, the rotor mass center G always lies below the rotational axis in the equilibrium due to the gravity (see Fig. 4.15a). That means the rotor will return from any position above the rotational axis to the equilibrium position because of its weight. Hence, one calls it static unbalance; obviously, it is easily seen that the rotor moves itself to the equilibrium position. On the contrary, in the couple unbalance the rotor does not return to the equilibrium position because the

(a)

It

(b)

m

polar mass-inertia axis

m

U

rotational axis

Ip

G

ε G

polar mass-inertia axis

U →

U = mε

g

Fig. 4.15 a Rotor static unbalance. b Rotor couple unbalance

Ip α rotational axis

4.2 Unbalance Whistle

65

rotor mass center G is always in the rotational axis. The rotor couple unbalance is only recognized in the rotating condition in which the unbalance moment acts upon the rotor despite without static unbalance, as shown in Fig. 4.15b. The static unbalance is caused by producing tolerances, such as large eccentricity at welding the turbine wheel on the rotor shaft, nonhomogeneous material, and residual unbalance after the shop balancing. Moreover, it is also induced by unbalance changes like thermal deformations, wear, and CW plastic deformations generated by rotor over-speeding, overheating the turbine wheel, and hard particle impact on the CW in low-pressure exhaust gas recirculation (EGR) turbocharging. On the contrary, the couple unbalance is created by a misalignment angle at welding turbine wheel in the rotor shaft, and unbalance changes, such as thermal and plastic deformations of the rotor due to rotor over-speeding, or loose screw-nut of the CW in the operation. In practice, both static and couple unbalances mostly occur in the turbine shaft manufacture, and CW assembly in the rotor shaft; these lead to the rotor dynamic unbalance. The dynamic unbalance is similar to the couple unbalance, but the rotor mass center G does not lie in the rotational axis (see Fig. 4.16). Therefore, the dynamic unbalance can be decomposed in the static and couple unbalances. In fact, neither the static unbalance nor the couple unbalance exists alone in automotive turbochargers; instead, the dynamic unbalance mostly occurs. The dynamic unbalance is generated in the turbine shaft manufacture at welding the turbine wheel in the rotor shaft. Additionally, the roundness and manufacturing quality of the screw-thread shape play a key role for causing the dynamic unbalances. The eccentricity e causes the static unbalance; the misalignment angle a induces the couple unbalance. Both unbalances are combined to the dynamic unbalance, as displayed in Fig. 4.16. The turbine shaft unbalance is the largest part of the entire rotor unbalance in the production. It contains the initial unbalance of the turbine wheel itself, eccentricity and misalignment at welding the turbine wheel in the rotor shaft.

It transverse mass-inertia axis rotational axis

O

G

GG

ε

O

α

polar mass-inertia axis

Ip Fig. 4.16 Rotor dynamic unbalance

66

4 Analyzing Root Causes of Noise

4.2.3 Unbalance Change The rotor dynamic change after a long-term operating time is considered as the unbalance change. Thermal and plastic deformations or location changes of the compressor and turbine wheels are to blame for the unbalance change. The maximum unbalance change is approximately 100 % of the delivery acceleration responses after the trim balancing according to experience. Therefore, the longterm changed unbalance of the rotor increases to nearly twice the delivery acceleration responses. About one-third of the unbalance change goes to the turbine shaft due to heat-related plastic deformations, internal stress relaxations at the welding interface between the rotor shaft and turbine wheel; two-third is caused by changing the CW positions in the axial, radial, and rotational directions. The CW unbalance change is generated by its new inappropriate positions of eccentricity, misalignment, rotating about the turbine shaft. These are caused by the wheel plastic deformation, the wheel dislocation in the radial direction due to overspeeding, and the wheel misalignment in the rotor shaft because of loose screw-nut at the compressor inlet. As a result, the delivery acceleration levels at the trim balancing must be reduced by at least one-half of the limit acceleration levels in which the unbalance whistle is inaudible in the vehicle cabin. The unbalance change ratio between the limit and delivery acceleration responses must be statistically determined by testing many turbochargers of the same type for the same vehicle type in a longterm operating time. If the ratio is chosen too large, many balancing steps are required for the trim balancing to keep the delivery acceleration levels small, leading to much time needed for the balancing process. Furthermore, this causes additional costs of lowered producing cycles, and cutting tools wear; therefore, more balancing machines are required for a large mass production. Otherwise, if the unbalance change ratio is selected too small, the long-term changed unbalance of the rotor exceeds the limit acceleration levels after a long-term operating time. As a result, the unbalance whistle is audible again in the cabin due to the unbalance change.

4.2.4 ICM Trim Balancing The ICM trim balancing is based on the Influence Coefficient Method, which has been developed by Goodman, Rieger, Lund and Tonnesen, Tessarzik, Badgley, and Anderson, etc., since 1961 [5, 6]. To improve the aeroacoustic quality (NVH) of automotive turbochargers, the trim balancing is normally carried out in two balancing planes at the screw-nut (plane 1*) and the hub surface between the blades near the CW outlet (plane 2*), as shown in Fig. 4.17. Based on the complex transfer impedance Z of the balancing system, i.e., the inversion of the complex dynamic stiffness coefficient KS of the system, including

4.2 Unbalance Whistle

67 →

y

→

→

(CW) →

→ U1

→

→

→

(TS)

U

→

U 2 ,b

→

U1,b

plane 1*

→

U = (U1 + U 2 ) = −U b = −(U1,b + U 2,b )

U2

→

Ω

plane 2*

Ub z=0

z =l

z

Fig. 4.17 ICM trim balancing at the balancing planes 1* and 2*

the turbocharger rotor and less influent balancing machine, the unbalance response at trim balancing is resulted from the complex transfer impedance and unbalance excitation force. At the trim balancing, note that to let the turbocharger possibly free vibrate in the test rig; do not clamp it on the test bank! The measured unbalance responses could be rotor vibration velocities or accelerations measured in the measuring planes. Theoretically, the measuring planes must be the same positions of the balancing planes 1* and 2* in the CW. However, the balancing planes rotate at high rotor speeds. It is cost intensive to measure the rotor unbalance responses in the rotating balancing planes in the mass production. In practice, they are computed from the measured rotor responses in the certain nonrotating position in the bearing housing or compressor and turbine casings at the trim balancing. This nonrotating position must be experimentally determined depending on turbocharger types. Due to the unbalance whistle, the rotor must be balanced at the high-speed trim balancing. Moreover, the rotor is flexible at high rotor speeds; it causes additional unbalances and unbalance moments. For this reason, the rotor should be balanced at high speeds in two balancing planes in the balancing plane 1* at the screw-nut and balancing plane 2* at the hub surface. A nozzle ring supplied by pressurized air at about 0.3 MPa absolute is used to accelerate the rotor up to the balancing speed of about 200,000 rpm. The pressurized air energy is transformed into kinetic energy in the nozzle ring with an air jet at the sonic speed acting upon the turbine wheel. Therefore, the turbine wheel is propelled by the nozzle ring at the wheel inlet. By twice timely differentiating the rotor amplitude, one obtains the acceleration amplitude measured in the trim balancing. jaj ¼ j€yðtÞj ¼ X2 jyðtÞj / N 2 jyðtÞj

ð4:35Þ

Note that the measured acceleration amplitude is proportional not only to the rotor amplitude but also to the rotor speed squared. It displays not exactly the rotor

68

4 Analyzing Root Causes of Noise

a ( sm2 )

a ∝ N2 limit acceleration response

aIII delivery acceleration response

aII

aII *

aI aI * 0

aIII *

NI

N II

N max

N (rpm)

Fig. 4.18 Limit and delivery acceleration levels at the ICM trim balancing

amplitude behavior, but the measured acceleration is strongly affected by the rotor speed squared N2, as shown in Fig. 4.18. As discussed in Sect. 5.5, the typical linear resonance does not exist in nonlinear rotordynamics, but only the limit cycle of the rotor response occurs at each rotor speed. The unbalance amplitude (1X) is relatively small compared to the asynchronous amplitudes in the rotor speed range. According to Eq. (4.35), the acceleration amplitude at high rotor speeds (N between NII and Nmax) could be higher than the acceleration amplitude at the quasi-resonance (Nres between NI and NII) although its rotor amplitude is less than the resonance rotor amplitude. Therefore, the radial bearing failure due to rub contact at the trim balancing happens at the first quasi-resonance instead of the maximum acceleration amplitude at high rotor speeds. Figure 4.18 and Table 4.1 give the delivery and limit acceleration levels for the rotor acceleration responses in the ICM trim balancing process.

4.3 Subsynchronous Constant Tone Inner oil whirl in the radial bearings of automotive turbochargers is to blame for the constant tone (howling), whose frequency ranges from 600 to 1,000 Hz in the human audible range. In rotating floating ring bearings, the inner oil whirl frequency order reduces from about 0.4 to 0.3X at increasing rotor speeds. Therefore,

Table 4.1 Delivery and limit acceleration levels in the balancing speed ranges Limit acceleration levels Balancing speed range Delivery acceleration levels a (m/s2) N (rpm) a* (m/s2) 0 \ N \ NI NI B N \ NII NII B N \ Nmax

aI* \ aI aII* \ aII aIII* \ aIII

aI aII aIII

4.3 Subsynchronous Constant Tone

69

the inner oil whirl frequency varies in small frequency range between 600 and 1,000 Hz compared to the rotor frequency; this frequency is usually considered as quasi-constant in the speed range of automotive turbochargers. The constant tone mostly occurs at the engine speed range between 1,500 and 3,500 rpm in second to fifth gear with middle to high loads. The inner oil whirl is caused by nonlinear characteristics of the oil film bearings. It is a subsynchronous vibration response whose frequency is less than the rotor frequency. As long as the damping force in the bearing is larger than the destabilizing force caused by the bearing cross-coupled stiffness coefficient, the journal whirling amplitude still remains small in quasi-stable condition. Hence, the constant tone is inaudible in the vehicle cabin. The journal begins whirling in the bearing inner oil film when the destabilizing force exceeds the damping force at increasing rotor speeds, leading to the self-excitation instability (subsynchronous inner oil whirl) which generates the constant tone. On the contrary, the outer oil whirl does not induce any noise due to its low frequency. Note that the noise intensity is proportional to its frequency squared. The outer oil whirl frequency is relatively low compared to the inner oil whirl frequency; therefore, the noise intensity is negligibly small. As a result, the outer oil whirl is inaudible in the car cabin although its whirl amplitude is much larger than the inner oil whirl amplitude. The problematic constant tone will be thoroughly dealt with in Sect. 6, in which the mutual effects between the constant tone and nonlinear rotordynamics with two oil film bearings are analyzed and discussed.

4.4 High-Order Harmonic Noise Ball bearings have no intrinsic damping due to very small clearances between the balls and races (see Fig. 4.20). As a result, they need an additional external damping to suppress the rotor amplitude responses at resonances and as well as to reduce airborne noises, such as the unbalance whistle and other aerodynamic noises induced by the ball bearings. In this case, a squeeze-film damper at the outer race is necessary for automotive turbochargers using ball and roller bearings. Contrary to the oil film bearings, the rolling element bearings do not induce the inner oil whirl; therefore, constant tone does not occur. However, besides the unbalance whistle (1X) the rolling element bearings have vibrations with multiple harmonic frequency orders of 2X, 3X, 4X, …, and modulation sidebands frequencies, as shown in Fig. 4.19. The inappropriate setup mechanism of the bearing cartridge in the bearing center housing is to blame for the high-order harmonic and side band noises [4]. Due to the unsuitable damper oil film between the bearing cartridge and bearing housing, high-order harmonic noises are generated at multiple harmonic frequencies. At small damper oil film thickness, induced noise is not suppressed enough and therefore transferred through the bearing center housing.

4 Analyzing Root Causes of Noise 5

High-order harmonic noises 4X side band noises 3X 2X 1X

a_LG_z 50k 75k 100k 2m 10m 50m

n/rpm a/m/(s^2) 1

4 3.5 3 2.5 2 1.5 1 0.5

Frequency Order ω / Ω

70

175k 200k 225k 2 5 10 20 100

Rotor speed Ω (rpm)

Fig. 4.19 Noise spectra of a ball bearing turbocharger (Courtesy BMTS)

However, high-order harmonic noises are induced at large damper oil film thickness. Therefore, the oil film damper set-up mechanism in the bearing housing must be thoroughly studied to improve the noise behavior of automotive turbochargers using the rolling element bearings.

4.5 Wear Noise Wear defects in the rolling-element bearings, such as defects in the inner and outer races, cage (retainer), and rolling elements (balls or rollers) generate different noise types that have asynchronous high-frequency orders and as well as sidebands frequencies due to frequency modulations. These wear noise frequencies are called the bearing defect-related frequencies [4]. Figure 4.20 shows the key elements and geometries of a ball bearing. The balls with diameter d are held in the cage (retainer); the pitch diameter Dp is defined by the maximum diametral distance between the centers of two opposite balls. Contact angle h is the angle between the line perpendicular to the shaft and the ball radius at the contact point in the outer race. The inner race is fixed to the shaft and rotates at the rotor speed N; the outer race is mounted in the cartridge supported by the damper oil film. The bearing defect-related frequencies are computed from the bearing geometries, number of balls, and rotor speed [2, 7]. Fundamental train frequency (ftf) is caused by the cage defects depending on the rotor speed N in rps.   N d 1 ftf ¼ cos h ð4:36Þ 2 Dp

4.5 Wear Noise

71 contact angle

θ outer race

outer race inner race

inner race

d balls

cage

Dp

cage (retainer)

balls

Fig. 4.20 Key elements of an angular contact ball bearing [4]

Ball passing frequency over defective inner race (bpfi) is generated when the Z balls or rollers pass over the defective inner race.   N d 1þ bpfi ¼ Z cos h ð4:37Þ 2 Dp Ball passing frequency over defective outer race (bpfo) is generated when the Z balls or rollers pass over the defective outer race.   N d 1 bpfo ¼ Z cos h ð4:38Þ 2 Dp Two-times ball spin frequency (2bsf) is induced by the defective balls or rollers spinning over the inner and outer races. " #  2 Dp d 2 2bsf ¼ N 1 cos h ð4:39Þ Dp d where d is Dp is h is Z is N is

the the the the the

diameter of balls or rollers; pitch diameter of the bearing; contact angle of the rolling elements; number of rolling elements; rotor speed in rps.

The bearing defect-related frequencies are displayed in Fig. 4.21 for a ball bearing containing eight balls with d = 3.2 mm, Dp = 12.2 mm, and h = 45°.

72

4 Analyzing Root Causes of Noise 18

bpfi

bpfi 16

Bearing frequency (kHz)

14 12

2bsf bpfo

2bsf

1X ftf

bpfo

10 8 6

1X

4

2 0 0,0E+00

ftf 1,0E+03

2,0E+03

3,0E+03

4,0E+03

Rotor speed N (rps)

Fig. 4.21 Bearing defect-related frequencies [4]

The frequencies induced bearing defects bpfi, 2bsf, and bpfo are supersynchronous (f [ 1X); the ftf frequency is subsynchronous (f \ 1X).

References 1. Cumpsty, N.A.: Compressor aerodynamics. Krieger Publishing Company, Malabar (2004) 2. Ehrich, F.: Handbook of rotordynamics. Krieger Publishing Company, Malabar (2004) 3. Greitzer, E.M.: The stability of pumping systems—the 1980 Freeman scholar lecture. J. Fluids. Eng. 103, 193–242 (1981) 4. Nguyen-Schäfer, H.: Rotordynamics of automotive turbochargers. Springer, Berlin-Heidelberg (2012) 5. Rieger, N.F.: Rotordynamics 2—problems in turbomachinery. CISM courses and lectures no 297. Springer, Wien, New York (1988) 6. Rieger, N.F.: Balancing of rigid and flexible rotors. U.S. DoD, Washington (1986) 7. Taylor, J., Kirkland, D.W.: The bearing analysis handbook. Vibration Consultants Inc., Tampa (2004) 8. Whitfield, A., Baines, N.C.: Design of radial turbomachines. Pearson Education, Longman Scientific and Technical, Harlow (1990)

Chapter 5

Computational Nonlinear Rotordynamics of Turbochargers

5.1 Vibration Equations of Automotive Turbocharger Rotors Figure 5.1 displays the rotor of an automotive turbocharger in the center housing and rotating assembly (CHRA) including the rotor shaft, compressor and turbine wheels, rotating floating ring bearings (RFRBs), seal rings, and thrust rings. All rotor components must be taken into account in the rotor dynamic computation to study the rotor vibration response, such as the frequency components in the Waterfall plot, rotor orbits in the phase plane, and response amplitudes in the time domain diagram. The continuum rotor vibration equations are generally discretized by two schemes: the finite element and transfer matrix methods. The finite element method (FEM), a discretization method of the continuum rotor structure, is based on the principle of D’ Alembert, where the sum of virtual work of all rotor elements equals zero. By discretization of the continuum rotor vibration equations, one divides the continuum rotor into a finite number of elements of the computational model that includes the concentrated masses, cylindrical elements, disks, and interfaces between the rotor, bearings, and seal rings. Therefore, the rotor vibration equations are derived and written in the discrete matrix Eq. (5.1). Another alternative method is the transfer matrix method (TMM) developed by Myklestad and Prohl, in which the rotor first element is connected with the last one by the overall transfer matrix that consists of the transfer matrices of all involved elements between the first and last rotor elements. The main advantage of the TMM is to spare the working memory compared to the FEM. The rotor vibration equations with N degrees of freedom (DOF), as shown in Fig. 5.1 are written in the discrete matrix equation. M€ x þ CSG x_ þ KS x ¼ fðtÞ

H. Nguyen-Schäfer, Aero and Vibroacoustics of Automotive Turbochargers, DOI: 10.1007/978-3-642-35070-2_5, Ó Springer-Verlag Berlin Heidelberg 2013

ð5:1Þ

73

74

5 Computational Nonlinear Rotordynamics of Turbochargers

turbine wheel

compressor wheel thrust rings

radial bearings

seal rings

rotor shaft

Fig. 5.1 Automotive turbocharger rotor setup in the CHRA (Courtesy BMTS)

where M = (N 9 N) is the mass matrix containing the rotor masses and inertia moments; CSG = (N 9 N) is the rotor damping coefficient and gyroscopic matrix; KS = (N 9 N) is the shaft stiffness coefficient matrix containing the diagonal and cross-coupled stiffness coefficients; x = (N 9 1) is the vibration rotor response vector including all degrees of freedom of two translational and two rotational displacements at each station of the element; f (t) = (N 9 1) is the vector of the unbalance forces and moments, and nonlinear bearing forces. To reduce the second-order to first-order equation system, the (2N 9 1) vector z of the vibration response is defined by         x x y x_ ¼ z ) z_ ¼ ¼ ð5:2Þ €x y x_ y_ where _ y_  € y  x; x: By substituting the vector z(t) into Eq. (5.1), 2N first-order vibration equations of z result in        x x_ 0 I 0 : þ ¼ _ tÞ € x_ x M1 ½UðX; tÞ þ FB ðx; x; M1 KS M1 CSG ð5:3Þ     0 I 0 , z_ ¼ :z þ _ tÞ M1 KS M1 CSG M1 ½UðX; tÞ þ FB ðx; x;

5.1 Vibration Equations of Automotive Turbocharger Rotors

75

where U is the unbalance force and moment; FB is the nonlinear bearing force computed by means of the impedance method that is resulted from solving the two-phase Reynolds lubrication equation, as given in Eq. (5.10). The coupled nonlinear vibration equation system (5.3) is written in the matrix equation of z. z_ ¼ Az þ bðz; X; tÞ

ð5:4aÞ

where A = (2N 9 2N) rotor mass, mass inertia moments, stiffness, and gyroscopic effect matrix; b = (2N 9 1) external force vector of unbalance and nonlinear bearing forces. Within



 0 I A¼ ; M1 KS M1 CSG   0 bðz; X; tÞ ¼ M1 ½UðXÞ þ FB ðz; tÞ

ð5:4bÞ ð5:4cÞ

The matrix A containing the matrices M, KS, and CSG is obviously nonsymmetric; i.e., A = AT where AT is the transposed matrix of A. For this reason, the bimodal method [1] is used to solve the first-order nonlinear matrix equation of z with a (2N 9 2 N) matrix A of N degrees of freedom.

5.2 Two-Phase Reynolds Lubrication Equation Figure 5.2 demonstrates a two-phase oil model in the bearing clearances in which the oil mixture volume Vmix consists of the volumes of gas bubbles VB and pure oil liquid Vliq. Obviously, one has Vmix = VB ? Vliq. The dynamic viscosity of the oil mixture is calculated from the oil liquid viscosity and gas bubble fraction according to [2].   1 gmix ¼ gliq ¼ gliq h ð5:5Þ 1þr where r  VVliqB is the bubble fraction in the oil liquid. The bearing filling grade h is defined by the ratio of the oil liquid to oil mixture volumes; it can be written in the bubble fraction r.

76

5 Computational Nonlinear Rotordynamics of Turbochargers

Vliq, ηliq

Vmix = Vliq + VB

VB, ηB ≈ 0 Vmix, ηmix

Oil mixture

Fig. 5.2 A two-phase oil mixture model

h

Vliq Vliq 1 ¼ ¼ Vmix Vliq þ VB 1 þ r

ð5:6Þ

Similarly, the oil mixture density is calculated from the oil liquid density and bearing filling grade. qmix ¼

mliq mliq mmix  ¼ Vmix Vliq þ VB Vliq ð1 þ VB =Vliq Þ   1 ¼ qliq h ¼ qliq 1þr

ð5:7Þ

The dynamic viscosity and density of the oil mixture given in Eqs. (5.5 and 5.7) depend only on the gas bubble fraction and are not affected by the dissolved gas in the oil liquid. There are two cases of the bearing filling grade: first, h = 1 for the pure oil liquid without air-releasing or cavitation; second, h = 0 for the oil condition at 100 % gas phase due to air release or cavitation in oil. However, the bearing filling grade lies between 0 and 1 (0 \ h B 1) in the most application cases. Having used the Henry-Dalton’s and Boyle-Mariotte’s laws, the gas bubble fraction is calculated at T and p by     VB Tp0 Tp0 r ¼ r0  av 1  Vliq T0 p T0 p   ð5:8Þ Tp0 ¼ ðr0 þ av Þ  av T0 p where av is the Bunsen’s coefficient chosen between 0.08 and 0.09 for the automotive mineral oils of ISO-VG 32-220 to 30 MPa. This Bunsen solubility coefficient is defined as the maximum dissolved gas volume per unit oil liquid volume in the equilibrium condition at the gas partial pressure of 101.3 kPa;

5.2 Two-Phase Reynolds Lubrication Equation

77

r0 is the initial gas bubble fraction in the oil liquid at the bearing inlet; T and p are the oil temperature and pressure in the oil film in the operating condition; T0 and p0 are the ambient or oil inlet temperature and pressure. Substituting Eq. (5.8) into Eq. (5.6), one obtains the bearing filling grade 0\ h 

Vliq 1    1 ¼ Vmix ð1  av Þ þ ðr0 þ av Þ Tp0 T0 p

ð5:9Þ

The Reynolds lubrication equation of two-phase flow lubricating oil is written in o qliq 3 op h ox gliq ox

!

o qliq 3 op þ h oz gliq oz

!



o o ¼ 6 ðUr þ Ub Þ ðqliq hhÞ þ 2 ðqliq hhÞ ox ot



ð5:10Þ where p(x, z) is the current oil film pressure in the bearing clearance; gliq(T) is the oil liquid viscosity computed from the Cameron and Vogel’s equation [4]; Ur and Ub are the circumferential velocities of the rotor and bearing ring; h(x) is the current oil-film thickness in the bearing clearance; h is the bearing filling grade, as given in Eq. (5.9).

5.3 Nonlinear Bearing Forces The bearing forces Fr and Ft are based on the rotating coordinate system (r, t) fixed to the journal (i.e., shaft inside the bearing). In fact, the bearing forces acting upon the journal in the inertial coordinate system (X1, X2) are required for the rotor Fig. 5.3 Coordinate transformation of the bearing force components

X2

t . γ ,γ ω Ob

Ft

F1

F2

X1

Oj

e

Fr FB α

γ

r

78

5 Computational Nonlinear Rotordynamics of Turbochargers

dynamic vibration equations. Therefore, the transformation from the rotating coordinates into the inertial coordinates is necessary to compute the bearing forces in the inertial coordinate system (X1, X2), as given in Fig. 5.3. The radial and tangential components of the bearing force FB acting on the journal, opposite to the journal force FJ, are resulted from integrating the oil pressure p(n, z) of Eq. (5.10) on the journal surface. 3 2 ZL Z2p 6 Fr ¼ R pðn; zÞ cos n dn:dz 7 7 6 7 6 0 0 7 6 FB ¼ 6 ð5:11Þ 7 ¼ FJ 7 6 ZL Z2p 7 6 4 Ft ¼ R pðn; zÞ sin n dn:dz 5 0

0

within

L is the inner bearing width; R is the journal or shaft radius; n is the position angle for the oil-film thickness where x = Rn(see Fig. 5.4). The nonlinear bearing forces in Eq. (5.11) are transformed into the inertial coordinate system (X1, X2):      F1 Fr sin c cos c FB ¼ ¼ ð5:12Þ F2 Ft  cos c sin c where F1, F2 are the force components in the inertial coordinates X1 and X2; Fr, Ft are the force components in the rotating coordinates r and t. Fig. 5.4 Bearing force characteristics

5.3 Nonlinear Bearing Forces

79

The nonlinear bearing force components are written in the inertial coordinates: ( F1 ¼ þFr sin c þ Ft cos c  f1 ðe; e_ ; c; c_ ; XÞ ð5:13Þ F2 ¼ Fr cos c þ Ft sin c  f2 ðe; e_ ; c; c_ ; XÞ where c is the angular position of the journal; e is the relative eccentricity of the journal (e = e/c); c is the bearing clearance; a is the attitude angle; X is the rotor speed.

5.4 Boundary Conditions of Nonlinear Rotordynamics In the following section, the computation of nonlinear rotordynamics for a typical automotive turbocharger using two separate rotating floating ring bearings has been carried out in the study case. The following characteristics are taken into account running the rotordynamic computational program Madyn 2000 [5], which is based on the extensionally developed module for nonlinear rotordynamics of high-speed turbochargers [4]: • • • • • • • • • • • • • • •

run-up simulation to very high rotor speeds; transient behavior of the rotor responses; gyroscopic effect of the flexible rotor; two-phase flows of the oil films in the bearing clearances [2, 3, 6]; unbalance forces and moments acting upon the rotor; damping effects of the seal rings; nonlinear bearing forces of the rotating floating ring bearings; synchronous and asynchronous vibration responses; rotating floating ring bearings with changing ring speed ratios; computing the ring speed ratios of the bearings; induced oil temperatures in the bearing clearances; heat convection in the oil films in the axial direction; heat conduction between the oil-films in the radial direction; various types of lubricating oil (SAE 0W30, 5W30, 10W40, 15W40); different pressures and temperatures of lubricating oil.

The computed rotor of an automotive turbocharger for passenger vehicles consists of the compressor and turbine wheels, rotor shaft, thrust rings, seal rings, and rotating floating ring bearings. One transforms this computed rotor into the discretized finite element model where the components of the rotor, such as the compressor and turbine wheels, rotor shaft, and radial bearings are discretized in a finite number of the concentrated mass points, disks, cylinder elements, and interfaces of the bearings and seal rings (see Fig. 5.5). The finite element model of the computed

80

5 Computational Nonlinear Rotordynamics of Turbochargers

compressor wheel

rotor shaft

turbine wheel

radial bearings

thrust rings

seal rings

section (i)

interface (i+1)

station (i)

station (i+1)

turbine wheel (TW)

compressor wheel (CW) thrust rings

U CW

S1 3

1 2

S2

S3

G1

U TW G2

radial bearings seal rings

Fig. 5.5 Computing rotor model and its finite element model [4]

rotor contains many sections that are connected by the stations at both ends of each section. By solving the two-phase Reynolds lubrication equation of two-oil-film bearings, the impedance table for the bearing forces is created for the bearing force computation. The nonlinear bearing forces in the bearing clearance are resulted from the corresponding bearing stiffness and damping coefficients by linearly interpolating the computed values of the bearing characteristics at the journal center position (e,c) and Sommerfeld number So in the impedance table. The nonlinear bearing forces contain two force components: the rotation force induced by the rotor speed X depends on the bearing force angle a and So (g,e,c); the damping force caused by the journal velocities e_ ; c_ depends on g; e; c; e_ ; c_ , as given in [4].

5.5 Computational Results of Turbochargers The nonlinear bearing forces acting upon the journal are computed at each iteration step based on the Sommerfeld number So by the impedance method. The coupled nonlinear vibration equations of the rotor are given in Eq. (5.3), in which

5.5 Computational Results of Turbochargers

81

we put the nonlinear bearing forces, as given in Eqs. (5.11 and 5.13) on the righthand side of Eq. (5.3). By using the bimodal method [1], we decouple the coupled nonlinear equations in the decoupled nonlinear equations. Then, the vibration responses are solved by a high-order Runge–Kutta scheme at each iteration step. Using the Discrete Fourier Transform (DFT) with FFT, the frequency spectrum of the rotor responses is displayed in the spectrogram (Waterfall plot). The rotor orbits at the station S1 of the compressor inlet and at the stations S2 and S3 of the journal loci in the radial bearings are computed in the runup simulation, as shown in Fig. 5.5. Two rotating floating ring bearings and two seal rings at each side support the computed rotor with a mass at about 150 g including the shaft with a diameter of 7 mm, compressor wheel, turbine shaft, and thrust rings. The relative inner diametral clearance of the bearing is initially about 3 9 10-3 due to the temperature difference between the inner and outer oil-films, the centrifugal force of the bearing ring, and the bearing ring expansion in the radial direction. Hence, the relative inner diametral clearance increases to a maximum about 4 9 10-3 during operation. The ratio of the outer and inner bearing clearances is varied by a factor between three and four. The maximum rotor speed is limited to nearly 225,000 rpm (X = 3,750 Hz). The compressor wheel and turbine shaft have residual unbalances of approximately 0.1 and 0.2 g.mm, respectively. The lubricating oil SAE 5W30 is used in the runup computation; the oil dynamic viscosity changes with the effective oilfilm temperatures in every iteration step according to the Cameron and Vogel’s equation [4]. The oil condition at the inlet is given at a relative pressure of 0.3 MPa and oil temperature of 90 °C at S2; 100 °C at S3. The computing time for the run-up simulation of about 20-hour-CPU is necessary on a PC with a dual-core AMD Opteron of 3 GHz with 8 GB RAM. This computing time is quite acceptable in the industry instead of a-few-week CPU in the case of instantaneously solving the Reynolds equation at each iteration step. The vibration mode of the flexible rotor in the run-up simulation is analyzed at the stations S2 and S3. Figure 5.6 displays the vibration modes of the rotor at various rotor speeds. At low rotor speeds, the rotor vibration has a conical mode. From the rotor speed of about 81,000 rpm (36 % relative rotor speed) – 135,000 rpm (60 % relative rotor speed), the rotor vibration changes from the conical to the cylindrical mode. At low rotor speeds up to nearly 45,000 rpm, the rotor orbits at S2 and S3 are mostly eccentric to each other (see Fig. 5.7), i.e., the mode shape of the rotor response is conical since the rotor is still rigid at low rotor speeds. At high rotor speeds from about 81,000 rpm, the bearing stiffness and damping coefficients increase, the conical mode of the rotor vibration tends toward cylindrical mode. Figure 5.7 shows that orbits at S2 and S3 are nearly concentric from the relative speed of 60 % up to the maximum rotor speed. The limit cycles of the rotor response at S2 and S3 have peak-to-peak amplitude of about 25 lm at 135,000 rpm and reduce to 15 lm at the maximum rotor speed of 225,000 rpm.

82

5 Computational Nonlinear Rotordynamics of Turbochargers

cylindrical mode @ 135,000 rpm

conical mode @ 80,000 rpm

rotor mode

cylindrical mode @ 170,000 rpm

cylindrical mode @ 225,000 rpm

Fig. 5.6 Vibration modes of the turbocharger rotor [4]

The unbalance amplitude (1X) is relatively small compared to the asynchronous amplitudes of the inner and outer oil whirls in the entire rotor speed range. In fact, the typical resonance due to unbalance does not exist in nonlinear rotordynamics (see Figs 5.8, 5.9a, and b), but only the limit cycle of the rotor response occurs at each rotor speed. The reason is the effective rotor stiffness coefficient depends not only on the shaft stiffness coefficient itself but also on the bearing stiffness coefficient. The rotor equivalent stiffness coefficient is called the nonsynchronous complex dynamic stiffness. Therefore, the bearing stiffness coefficient is dependent on the effective oil pressure, temperature, rotor speed, bearing ring speed, and journal eccentricity as well. At large rotor deflections, the journal eccentricity increases in the bearing clearance; the oil-film thickness reduces in the bearing clearance to generate the large bearing force. As a result, the bearing stiffness and damping coefficients increase with the rotor deflection until the journal orbit stabilizes after the Hopf bifurcation point in the limit cycle at each rotor speed [4]. In this case, no typical resonance like in linear rotordynamics occurs in the entire rotor speed range, as shown in the Waterfall plot (see Fig. 5.8). In practice, one plots Waterfall plot in a two-dimensional plot (modified Waterfall plot), in which the amplitudes of the vibrations are displayed in color or grayscale in the frequency order spectra, as shown in Fig. 5.9a. Instead of the whirl frequency x, the frequency order x/X versus rotor speed X is used in the modified

5.5 Computational Results of Turbochargers

90,000 rpm (40%)

45,000 rpm (20%) S2

83

limit cycle of the total response S2

S3

S3

225,000 rpm (100%)

135,000 rpm (60%)

S2

S2

S3

S3

Fig. 5.7 Orbit of the journal at the stations S2 and S3 [4]

Waterfall plot. Its main advantage is it is easier to analyze the frequency order spectra of the synchronous and asynchronous vibrations at the rotor speed. The frequency jump of the inner oil whirl takes place at nearly 30 % of relative rotor speed, where the conical vibration mode changes into the cylindrical mode. Figure 5.9a and b shows the comparison between the computation of rotordynamics and measurement results in Waterfall plots. Compared to the measurement, the computed inner oil whirl takes place a little later at the relative rotor speed of about 30 % (67,500 rpm) instead of at nearly 27 % (62,000 rpm) as in the measurement. The frequency order of the computed inner oil whirl (inner OW) begins at a lower frequency order of 0.4X compared to about 0.55X in the measurement and ends at the frequency order of 0.2X, nearly the same as in the measurement results. The low frequency order of the inner oil whirl is caused by the higher

84

unbalance (1X)

inner oil whirl

Amplitude(µm)

Relative rotor speed N/N

max

(%)

outer oil whirl

5 Computational Nonlinear Rotordynamics of Turbochargers

N max = 225,000 rpm

Whirl frequency (Hz)

Fig. 5.8 Waterfall plot of the journal at the station S1 [4]

bearing ring speed in the measurement than in the computation. The unknown heat flow transferred from the bearing housing to the outer oil-film of the bearing has not been taken into account in the computation. For this reason, the outer oil-film temperature increases in the measurement compared to the computation; therefore, the ring speed ratio of the bearing is higher, leading to the higher frequency order of the inner oil whirl. In fact, the residual unbalance vector directions of the compressor and turbine wheels are unknown after the trim balancing in the measurement. For this reason, the resulting residual unbalance may be smaller in the computation. Thus, the frequency jump of the inner oil whirl occurs at a rotor speed of about 67,500 rpm a little later than in the measurement at 62,000 rpm. Similarly, the computed frequency order of the outer oil whirl (outer OW) is a little lower than the frequency order as in the measurement due to the reduced bearing ring speed, especially at low rotor speeds. In the rotating floating ring bearing case, the journal moves relatively to the bearing ring at every rotor speed; i.e., the journal motion pushes the bearing ring in the radial direction. In fact, it is very difficult to measure the relative displacements between the journal and bearing ring during the rotation. Hence, the rotordynamic computation is a useful tool to determine the relative displacements and the current oil-film thickness in the rotating floating ring bearings at any rotor speed. Figure 5.10 displays the relative displacements between the journal and bearing ring at different rotor speeds.

5.5 Computational Results of Turbochargers

(a)

85

computed

Rotor amplitude (µm)

Frequency order ω ⁄ Ω

unbalance (1X)

inner oil whirl jump outer oil whirl Nmax = 225,000 rpm Relative rotor speed (%)

(b)

measured 1.1 1 0.9

unbalance (1X)

0.7 0.6

jump

0.5

inner OW

0.4

Frequency order

0.8

0.3

outer OW

0.2 dis_z 0.1

50k 1µ

75k 10µ

100k

125k 0.1m

n/rpm 150k length/mm

175k 10m

200k 0.1

225k 1

Rotor speed N (rpm)

Fig. 5.9 a and b Waterfall plots of the journal vibration at S1 [4]

At the beginning, the journal and bearing ring are nearly concentric to each other since the rotor mass is very small at nearly 150 g. If the relative displacement is positive, the journal moves closer to the bearing ring because the journal displacement is larger than the ring displacement. In the other case, the journal moves away from the bearing ring. The current oil-film thickness is resulted from the bearing clearance and relative displacement. When the oil-film thickness is larger than the limit oil-film thickness [4], the lubrication regime is fully hydrodynamic and outside the mixed and boundary lubrications; little or no wear occurs in the bearing. The computed results show that the maximum relative displacement is about 9 lm in the limit cycle for the entire rotor speed range. It indicates that the minimum oil-film thickness in the inner bearing clearance is nearly 5 lm with the

86

5 Computational Nonlinear Rotordynamics of Turbochargers 80,000 rpm (35%)

S2

S3

135,000 rpm (60%)

S2

225,000 rpm (100%)

170,000 rpm (75%)

S2

S3

S2

S3

S3

Fig. 5.10 Waterfall plots of the relative displacements between the journal and bearing ring at S2 and S3 [4]

inner radial clearance of 14 lm. The minimum oil-film thickness of 5 lm in the bearing clearance is larger than the limit oil-film thickness; hence, no wear occurs in the bearing.

References 1. Nordmann, R.: Schwingungsberechnung von nichtkonservativen rotoren mit hilfe von linksund rechts-eigenvektoren (in German), VDI Bericht 269 (1976) 2. Bartel, D.: Simulation von tribosystem (in German). Springer-Teubner, (2010) 3. Nguyen-Schäfer, H.: Nonlinear rotordynamics computation of automotive turbochargers using rotating floating ring bearings at high rotor speeds. 10th SIRM International Conference, Berlin (2013) 4. Nguyen-Schäfer, H.: Rotordynamics of automotive turbochargers. Springer, Berlin-Heidelberg (2012)

References

87

5. Schmied, J: Program Madyn 2000 for computation of rotordynamics, Delta J.S., Zurich (2011) 6. Nguyen-Schäfer, H., Sprafke, P.: Numerical study on interaction effects of the bubbles induced by air-release and cavitation in hydraulic systems, tenth bath international fluid power workshop bath. UK Research Studies Press Ltd, England (1997)

Chapter 6

Subsynchronous Constant Tone

Constant tone is induced by the inner oil whirl in hydrodynamic radial bearings. The inner oil whirl frequency order in the rotating floating ring bearings reduces from 0.4X to about 0.3X at increasing rotor speeds, at which the inner oil film temperature increases, leading to reduced oil viscosity. Therefore, the inner oil whirl frequency varies in a narrow frequency band from 600 to 1,000 Hz of the human audible range. This frequency is usually considered in automotive turbochargers as quasi-constant compared to the rotor frequency; hence, its noise is called constant tone. The constant tone occurs at the engine speed range between 1,500 and 3,500 rpm in second to fifth gears with middle to high loads. Figure 6.1 shows the unbalance whistle and constant tone acceleration frequency spectra in the Waterfall plot. Oil whirl is the self-excited instability with a subsynchronous frequency that usually occurs in automotive turbochargers. The oil whirling in the bearing clearance causes the oil whirl; therefore, it has nothing to do with the rotor unbalance. This means that the oil whirl happens in the bearing without precondition whether the rotor is well balanced or unbalanced. However, a strongly unbalanced rotor could reduce the oil whirl amplitude where the rotor is close to the bearing wall. The constant tone level is measured by an acceleration sensor at the bearing center housing, as displayed in Fig. 6.1. The constant tone is a subsynchronous vibration response; i.e., its whirl frequency is lower than the rotor frequency. For this reason, it is a subsynchronous noise type. As long as the damping force in the bearing is larger than the destabilizing force caused by the bearing cross-coupled stiffness coefficient, the journal orbit is in a quasi-stable condition. Hence, the constant tone level is inaudible in the vehicle cabin. The journal begins whirling in the bearing when the destabilizing force exceeds the damping force at increasing rotor speeds, leading to the self-excitation instability of the subsynchronous inner oil whirl, which generates the constant tone. Note that the noise intensity is proportional to the whirl frequency squared. The outer oil whirl frequency is much lower than the inner oil whirl frequency; therefore, the outer oil whirl noise intensity is negligibly small. As a result, the outer oil whirl noise is inaudible although its whirl amplitude is much larger than the inner oil whirl amplitude. H. Nguyen-Schäfer, Aero and Vibroacoustics of Automotive Turbochargers, DOI: 10.1007/978-3-642-35070-2_6, Ó Springer-Verlag Berlin Heidelberg 2013

89

90

6 Subsynchronous Constant Tone

10k 5k unbalance whistle (1X) f/Hz 2k 1k

ω / kHz constant tone

500 200

a_Ölab_z

75k 100k 125k n/rpm150k 2m 5m 10m20m 50m0.1 a/m/(s^2) 1 2

175k 200k 5 10 20 100

Ω / rpm

Fig. 6.1 Acceleration frequency spectra in Waterfall plot measured at the bearing center housing [4]

6.1 Transmitting Way of Airborne Noise Synchronous and asynchronous (super- and subsynchronous) noises induced in automotive turbochargers are caused by compressor aerodynamics of compressor and turbine wheels rotation, and rotordynamics of rotor unbalance and inner oil whirl in the radial oil film bearings. Induced noises transmit through the bearing oil films, bearing center housing, compressor housing, air filter, charge-air intercooler, exhaust-gas manifold, exhaust-gas system (catalytic converter, particle diesel filter, and muffler), and car frame to the vehicle cabin, as shown in Fig. 6.2. On the transmitting ways, they excite the periphery components neighboring the turbocharger. When the noise source frequencies equal the component eigenfrequencies, resonances take place with high response amplitudes in low damping cases (see Chap. 7). The vibration responses of the bearing center housing, air filter, charge-air intercooler, and exhaust-gas system additionally emit airborne noises to the car cabin and environment. Airborne noises are uncomfortable and undesirable for driver and passengers; therefore, they must be reduced as much as possible in passenger vehicles. The airborne noise level individually depends on the vehicle types because their eigenfrequencies, damping and stiffness coefficients of the periphery components, and car frame are non-identical to each other; therefore, these lead to different noise responses. As a result, the vehicle characteristics play an important role in the airborne noise level of automotive turbochargers. Note that some airborne noises of a turbocharger are inaudible in one vehicle type, but they could be audible in another vehicle type. For this reason, the airborne noise behavior of turbochargers must be individually proofed for each vehicle type in their applications.

6.2 Constant Tone Root Causes

91

Noise sources

Exhaust Gas Turbocharger

• Oil films of bearing • Bearing center housing • Compressor housing • Charge air intercoolers • Intake air filter • Exhaust gas system (Catalyzer, DPF, Muffler)

Airborne noise Car frame

Fig. 6.2 Transmitting way of airborne noise

6.2 Constant Tone Root Causes The inner oil whirl in the radial bearing is to blame for the constant tone, as earlier discussed. To better understand the constant tone, the inner oil whirl root causes must be analyzed. In the following section, the dynamic force model for the journal and flow model for rotating floating ring bearings are used to study the constant tone. Such analytical models could answer the questions, such as why and when the inner oil whirl takes place in the bearing, which influential parameters the constant tone depend on, and how the constant tone could be reduced in passenger vehicles. Eventually, measures of constant tone reduction are discussed. At first, the model of dynamic forces acting upon the journal in the rotating coordinate system (r, t) fixed to the journal is displayed in Fig. 6.3. In the radial direction r, the journal stiffness force Ke, bearing cross-coupled damping force jcc e_ , inertial force m€e, and centripetal force mre x2 ejxt act upon the journal. The destabilizing force jkc e caused by the cross-coupled stiffness coefficient kc of the oil film and the damping force of the bearing c_e act upon the journal in the tangential direction t. The inner oil whirl does not occur as long as the damping force is larger than the destabilizing force, as shown in Fig. 6.3. Thus, the stability condition for the inner oil whirl is formulated in the tangential direction: c_eðtÞ [ jkc eðtÞ

ð6:1Þ

92

6 Subsynchronous Constant Tone

OB

x e = (O O B

J

)

whirl orbit

ω

t

ωt − Ke

jcc e

− me

jk e

m

c

destabilizing force

Ω OJ ωt

mrε ω 2e j

− ce damping force

journal

r

y Fig. 6.3 Acting forces on the journal of an oil film bearing

where e_ ¼ jx e is the journal eccentricity velocity; x is the whirl speed; kc ¼ ckX is the cross-coupled stiffness coefficient of the oil-film bearing within X is the rotor speed; k is defined by the average fluid velocity ratio according to [1]. The average fluid velocity ratio k versus the journal relative eccentricity e is displayed in Fig. 6.3. By using the rotating floating ring bearing with two oil films, two subsynchronous vibration responses of the inner and outer oil whirls induce in the bearing next to the rotor unbalance harmonic vibration response. The constant tone is induced by the inner oil whirl occurring in the radial bearing oil film. At increasing rotor speeds, the journal begins whirling in the inner bearing clearance when the damping force is less than the destabilizing force. In this case, the non-dissipated whirling energy excites the rotor to the self-excited unstable response (inner oil whirl) at a whirl frequency x. The stability condition for the inner oil whirl can be derived from Eq. (6.1) and is written in x [ kX:

ð6:2Þ

6.2 Constant Tone Root Causes

93

Equation (6.2) indicates that the rotor remains stable as long as the whirl angular velocity x is larger than kX, in which the feeding kinetic energy of the destabilizing force is mostly dissipated by the damping energy in the radial bearing. At high rotor speeds, the effective temperature of the inner oil film rises due to bearing friction, leading to the oil viscosity decrease in the inner bearing clearance and the reduction of dissipated damping energy as well. As soon as x \ kX, the rotor becomes unstable and begins to whirl in the whirl orbit at the whirl frequency x (see Fig. 6.3). In the case of using the rotating floating ring bearings, the stability condition for the inner oil whirl results in x [ kðX þ XR Þ

) x [ kX ð1 þ RSRÞ

ð6:3Þ

where RSR = XR/X, is the ring speed ratio of the bearing; XR is the ring angular speed; X is the angular rotor speed. In the case of the semi-floating ring or fixed bearings, RSR equals zero; thus Eq. (6.3) becomes the earlier stability condition (q.e.d.) given in Eq. (6.2). On the one hand, the destabilizing force increases with rotor speed since the bearing cross-coupled stiffness coefficient is proportional to the rotor speed. On the other hand, the damping force in the bearing decreases with rotor speed because the inner oil film temperature in the bearing increases due to bearing fiction; therefore, its viscosity strongly reduces at increasing oil temperatures. As a result, the damping force becomes small and underlies the destabilizing force in the bearing at high rotor speeds. This causes the inner oil whirl, which is a subsynchronous forward whirl; its precession has the same rotor velocity direction. The threshold of instability of the rotor is studied and analyzed in the following subsection. Having applied Newton’s second law to the rotor, the rotor vibration equation is written in the rotating coordinate system (r, t) fixed to the journal. m€e þ ðc  jcc Þ_e þ ðK  jkc Þe ¼ mre x2 ejxt where m is the rotor mass; j is the imaginary unit (j2 = - 1); e is the journal eccentricity; c is the bearing diagonal damping coefficient; cc is the bearing cross-coupled damping coefficient K is the effective stiffness coefficient of the rotor-bearing system; kc = ckX is the bearing cross-coupled stiffness coefficient; x is the whirl speed; re is the rotor unbalance radius.

ð6:4Þ

94

6 Subsynchronous Constant Tone

The rotor response is assumed as an exponential function of the eigenvalue s and time t. eðtÞ ¼ Aest

ð6:5Þ

where s is the complex eigenvalue including the real part a and the imaginary part xn. s ¼ a  jxn

ð6:6Þ

By twice differentiating Eq. (6.5), one achieves e_ ¼ sAest ¼ seðtÞ; €e ¼ s2 Aest ¼ s2 eðtÞ

ð6:7Þ

Substituting Eqs. (6.5) and (6.7) into the homogeneous Eq. (6.4), the rotor response characteristic equation is given by DðsÞ  ms2 þ ðc  jcc Þs þ ðK  jckXÞ ¼ 0

ð6:8Þ

The rotor stability condition is satisfied if the real part of the complex eigenvalue s is always negative. The real part a is calculated from the characteristic equation (see Appendix C). a¼ within E

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 12 c 1 h  pffiffiffi E þ E2 þ F 2 \0 2m 2

 2  K c  c2c  ; m 4m2

F

c cc  kX  m 2m

ð6:9Þ

ð6:10Þ

By solving Eq. (6.9), one obtains the stability condition for the inner oil whirl. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cc K  cc 2 þ þ [ kX ð6:11Þ x ¼ m 2m 2m

At low rotor speeds, the rotor stability condition in Eq. (6.11) is satisfied because the rotor orbit is relatively small and quasi-stable. As the rotor speed increases, the inner oil film temperature rises due to bearing friction, which leads to decrease of the cross-coupled damping and bearing stiffness coefficients. This causes the reduced whirl frequency x given in Eq. (6.11). Therefore, the rotor stability condition is no longer satisfied because the whirl frequency x underlies kX at high rotor speeds. In this case, the oil whirl orbit becomes unstable, leading to the self-excited oil whirl instability, which corresponds to the constant tone. The threshold of instability frequency is given by setting the real part a equal zero, where the oil whirl instability begins.

6.2 Constant Tone Root Causes

95

1 cc þ Xth ¼ k 2m

! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K  cc  2 þ m 2m

ð6:12Þ

Oil whip, a self-excited subsynchronous instability occurs at the rotor speed in which the oil whirl frequency x equals the first bending critical frequency Xcr. The oil whip is very dangerous and destructive because the rotor defection continuously exponentially increases with time in the bearing clearance even at a constant rotor speed, leading to the bearing damage due to wear or seizure of the journal in the bearings.

6.3 Constant Tone Analysis In the following section, influential parameters of the constant tone are studied in the flow model of a two-oil-film radial bearing. The rotor angular speed is defined by X; the ring speed of the rotating floating ring bearing by XR. At the journal eccentricity e, the inlet bearing clearance is obviously equal to (c + e); the outlet bearing clearance equals (c - e), as displayed in Fig. 6.4. Having considered the inner bearing convergent wedge, the oil volumetric flow rate at the inlet is computed. Q_ in ¼ vin Ain ¼ ki ½XR þ XR ðR þ cÞðc þ eÞLi

ð6:13Þ

outer oil film

rotatingfloating bearing ring

cavitating inner oil film journal

vin = λ i [ΩR + Ω R ( R + c)]

ωi O

B

bearing housing

vi

e O J Ω

journal orbit

ΩR

R

convergent wedge

Fig. 6.4 Oil velocity profiles in the bearing clearance

vout = λ i [ΩR + Ω R ( R + c)]

96

6 Subsynchronous Constant Tone linear velocity profile

ω λ= (Ω + Ω R )

Ω

journal

λ ≈ 0.5

ω ΩR

bearing ring

λ 0 ≈ 0.5

concave velocity profile

Ω

λ < 0.5

ω ΩR

λ = f (ε )

bearing contact

(ω = 0) λ =0

0 0

1

ε=e c

Fig. 6.5 Average fluid velocity ratio k versus journal relative eccentricity e

where ki is the average fluid velocity ratio, as given in Fig. 6.5; c is the radial inner bearing clearance; e is the journal eccentricity; Li is the inner bearing width; R is the journal radius. Analogous to Eq. (6.13), the oil volumetric flow rate at the outlet results in Q_ out ¼ vout Aout ¼ ki ½XR þ XR ðR þ cÞðc  eÞLi

ð6:14Þ

According to Eqs. (6.13) and (6.14), the oil inflow rate into the convergent wedge is more than the oil outflow rate. However, the supply oil can only flow in the axial direction toward the bearing ends, but the axial oil flow rate is very small due to the small inner bearing clearance. As a result, the journal has to move at the whirl velocity vi tangent to the whirl orbit in order to satisfy the mass balance in the inner bearing clearance. The journal whirl frequency xi corresponds to vi, as shown in Fig. 6.4. The oil surplus volumetric flow rate in the convergent wedge is given by DQ_ ¼ Q_ in  Q_ out  2RLi eki ðX þ XR Þ ¼ 2RLi eki Xð1 þ RSRÞ

ð6:15Þ

where RSR = XR/X is the ring speed ratio of the bearing. The ring speed ratio depends not only on the bearing geometries but also on the effective oil temperatures in the bearing clearances during operation (see Sect. 6.4).

6.3 Constant Tone Analysis

97

At the journal whirling, the removed oil volumetric flow rate (i.e. oil whirling volumetric flow rate) results in Q_ w ¼ vi Ai ¼ ðexi Þð2RLi Þ ¼ 2RLi exi

ð6:16Þ

where Ai is the displaced journal cross-sectional area; A i = 2RLi xi is the journal whirl angular frequency. Having used the continuity equation in the inner bearing convergent wedge, X Q_ i ¼ DQ_  Q_ w  Q_ ax ¼ 0 ð6:17Þ i

The oil whirling volumetric flow rate in Eq. (6.15) is rewritten in Q_ w ¼ 2RLi exi

¼ 2RLi eki Xð1 þ RSRÞ  Q_ ax

ð6:18Þ

The inner oil whirl amplitude is proportional to the oil whirling volumetric flow rate given in Eq. (6.18). The journal forward whirl speed is resulted from Eq. (6.18) in xi ¼ ki Xð1 þ RSRÞ 

Q_ ax 2RLi e

ð6:19Þ

Since the oil flow rate in the axial direction is negligibly small, the inner oil whirl frequency near the bearing center at ki & 0.5 becomes 1 2

xi ¼ ki Xð1 þ RSRÞ  Xð1 þ RSRÞ

ð6:20Þ

In the case of semi-floating ring or fixed bearings at RSR = 0, the constant tone frequency is nearly one-half rotor frequency (xi& 0.47X). It is usually called the half-frequency oil whirl. The average fluid velocity ratio of rotating floating ring bearings is qualitatively displayed in Fig. 6.5.

6.4 Calculating Ring Speed Ratio Floating ring bearings are usually applied to automotive turbochargers to reduce airborne noise and rotor amplitudes at resonances. There are two common types of floating ring bearings: semi-floating (SFRB) and rotating-floating ring bearings (RFRB). The common characteristic of both floating ring bearings is having two oil films: inner and outer oil films. However, the semi-floating bearing ring is non-rotating but floating; the rotating-floating bearing ring is rotating and floating at once.

98

6 Subsynchronous Constant Tone

In order to reduce the bearing friction, the rotating floating ring bearings (RFRB) are used in automotive turbochargers. Due to reduction of the bearing friction at low rotor speeds, the transient response of turbochargers is improved in low-end torque. Figure 6.2 shows the CHRA setup with the rotating floating ring bearings. Lubricating oil at the pressure pi from the inlet flows through the outer bearing clearance in the circumferential and axial directions and leaves the bearing ends at the ambient pressure p0. By means of oil holes in the bearing ring, lubricating oil enters the inner bearing clearance, flows in the circumferential and axial directions, and leaves the bearing ends at the ambient pressure p0. The volumetric oil flow rate depends on the oil inlet temperature, pressure drop between pi and po, rotor speed, as well as the rotating speed of the bearing ring. The inner oil film has to carry the rotor to balance external forces acting upon it. Therefore, its inner bearing radial clearance c1 is relatively small to increase the bearing stiffness coefficients. On the contrary, the outer oil film provides the rotor with large damping effects to reduce the rotor deflection at resonances, and suppress the rotordynamic airborne noises of the unbalance whistle and constant tone. Flow characteristics of the oil films are taken into account in computing the friction power in the bearing (see Fig. 6.6). The bearing friction power produced in the rotor shaft is given by   Z oU Uj Aj Uj  gi Aj Uj Pf ¼ Ff Uj ¼ Uj s dA ¼ gi ð6:21Þ oh h A

where gi is the inner oil film dynamic viscosity; oU=oh is the oil film velocity gradient; h is the oil-film thickness;

rotating floating bearing ring

outer oil film inner oil film

U R ≈ ΩR R

journal

U j = ΩR ω OB

e OJ

Ω ΩR R0

Fig. 6.6 Flow velocities in the oil films

R

U R0 ≈ Ω R R0

6.4 Calculating Ring Speed Ratio

99

Aj is the journal frictional area; Uj is the journal circumferential velocity. In order to reduce the bearing friction power, the velocity gradient is decreased when the bearing ring rotates at a circumferential velocity UR at the inner bearing clearance. Hence, the bearing friction power is reduced to Pf  gi

ðUj  UR Þ A j Uj h

ð6:22Þ

According to Eqs. (6.21) and (6.22), the relative reduction of the bearing friction power is computed. Pf DPf Pf  Pf UR XR  RSR ¼1 ¼ ¼  Pf Pf Pf Uj X

ð6:23Þ

where XR is the ring angular velocity (XR \ X); X is the rotor angular velocity. Equation (6.23) indicates that the friction power reduction is proportional to the ring speed ratio RSR. Therefore, the higher the ring speed ratio, the larger the bearing friction power decreases. The Petroff’s law resulting from the Reynolds lubrication equation computes the torque acting upon the journal in the semi-floating ring bearing. Z Z oU 2pgR3 LX M ¼ R s dA ¼ R g dA ¼ ð6:24Þ oh c A

A

Having applied Eq. (6.24) to the rotating floating ring bearing, one obtains the driving torque acting on the bearing ring inner side, Mi ¼

2pgi R3i Li ðX  XR Þ c1

ð6:25Þ

and the friction torque on the bearing ring outer side Mo ¼ 

2pgo R3o Lo XR c2

ð6:26Þ

Having used the angular momentum law, the ring speed ratio RSR is resulted at the steady-state condition ð€ h ¼ 0Þ. X h¼0 M ¼ Mi þ M o ¼ Ip € Thus,

RSRsteady 

XR  X

1þ

go ðTo Þ gi ðTi Þ

1    3 Lo Li

c1 c2

Do Di

ð6:27Þ

100

6 Subsynchronous Constant Tone

pi RFRB

oil holes Lo c2 shaft

Do

c1 Li Di

Ω po

Fig. 6.7 Rotating floating ring bearing geometries

where gi, go are the dynamic viscosity of the inner and outer oil films; Li, Lo are the inner and outer bearing widths (see Fig. 6.7); c1, c2 are the inner and outer radial bearing clearances; Di, Do are the inner and outer bearing diameters; Ip is the polar mass inertia moment of the rotor. When the inner and outer relative eccentricities e1 and e2 are taken into account, the modified steady-state RSR* becomes according to [2]. RSRsteady 

XR  X

1   pffiffiffiffiffiffiffi2ffi 3 c1 1e1 ðTo Þ Lo pffiffiffiffiffiffiffi2ffi DDo 1 þ ggo ðT Þ L i

i

1e2

c2

i

ð6:28Þ

i

In the unsteady state at € h 6¼ 0; the ring speed ratio in Eqs. (6.27) and (6.28) is modified according to [3]. RSRunsteady ¼ RSRsteady 

Ip €h 2pX



gi R3i Li c1

þ

go R3o Lo c2



ð6:29Þ

Because the inner clearance is much smaller than the outer clearance, the inner oil-film temperature is higher than the outer oil-film temperature at increasing rotor speeds due to the bearing friction. Hence, the oil viscosity ratio between the outer and inner oil films increases (i.e., go [[ gi), leading to the ring speed ratio decrease. The computed ring speed ratio versus rotor speed at different oil inlet temperatures of the rotating floating ring bearing is displayed in Fig. 6.8. In the computation, the effective temperatures Tb,i and Tb,o of the inner and outer oil films

6.4 Calculating Ring Speed Ratio

101

Ring speed ratio RSR [-]

Oil inlet temperature Tin

Rotor speed N [rps] Fig. 6.8 Computed ring speed ratio RSRsteady versus rotor speeds

are induced only by the bearing friction without the heat conduction from the bearing center housing to the oil films. In fact, the heat conduction from the bearing center housing increases the outer oil film temperature. Therefore, the ring speed ratio at the turbine side is normally quite larger than the ring speed ratio at the compressor side. The ring speed ratio varies between 15 and 30 % at low rotor speeds to about 90,000 rpm (1,500 cps) at the oil inlet temperature of 90 °C. Hence, the rotor friction power of the rotating floating ring bearing could be reduced by 20–30 % compared to the semi-floating ring bearing in low-end torque (LET), leading to improving the turbocharger’s transient behavior. The higher the rotor speed, the higher the effective inner oil film temperature is, leading to the ring speed reduction. At high inlet oil temperatures, the dynamic viscosity ratio of both outer and inner oil films does not change so much; therefore, the ring speed ratio reduces more slowly and remains nearly constant at higher rotor speeds. At decreasing inlet oil temperatures, the bearing friction increases in the inner oil film, leading to high inner oil film temperatures. As a result, the ring speed is lowered as the inlet oil temperature decreases. Generally, the ring speed ratio decreases with the rotor speed at a constant oil inlet temperature and increases with the oil inlet temperature at a constant rotor speed, as shown in Fig. 6.8. The rotating floating ring bearings have two main advantages: first, the bearing friction reduction, especially in low-end torque; second, the large damping effect due to the large outer oil film. However, they need large oil volumetric flow rate and the rotor orbit is larger than the orbit of the semi-floating ring bearings because of having two oil films. If we consider the unsteady state in the rotor run-up phase, the ring speed ratio is very low at small rotor speeds and high inlet oil temperatures because the inertia term of the ring speed ratio in Eq. (6.29) is significantly large, especially at high

102

6 Subsynchronous Constant Tone

ω unbalance whistle (1X)

Nmax

run-up

slowdown unbalance whistle @ run-up

∼ 1000 Hz

unbalance whistle @ slowdown

400 Hz…600 Hz pre constant tone @ run-up

post constant tone @ slowdown time t

Fig. 6.9 Induced noises in the run-up and slowdown phases

accelerations. At increasing rotor speeds, the inertia term becomes smaller; it leads to the ring speed ratio increase. Then, the ring speed ratio is steady state at further increasing rotor speeds.

6.5 Pre and Post Constant Tone In the run-up phase, turbocharger rotor is accelerated from the idle condition to maximum rotor speed, as displayed in Fig. 6.9. The oil film temperature in the inner bearing is relatively low and nearly equals the outer oil film temperature at small rotor speeds. Therefore, the ring speed ratio given in Eq. (6.29) increases slowly during the rotor acceleration. According to Eq. (6.20), the inner oil whirl frequency is nearly proportional to the product of the rotor speed and ring speed ratio. As a result, the pre constant tone frequency varies from 400 to 600 Hz in the audible frequency range. At further increasing rotor speeds to the maximum speed, the inner oil film temperature rises higher than the outer oil film temperature because the viscous friction in the inner oil film is larger than the friction in the outer oil film, leading to viscosity reduction in the inner oil film. Hence, the ring speed ratio reduces with rotor speeds according to Eqs. (6.28) and (6.29). On the one hand, the inner oil whirl frequency decreases with reducing ring speed ratios at high rotor speeds. On the other hand, the inner oil whirl frequency is proportional to the rotor speed, as shown in Eq. (6.20). As a result, the constant tone frequency increases much slowly at high rotor speeds and therefore ranges from 600 to 1,000 Hz. Thus, the constant tone is considered quasi-constant compared to the rotor frequency in the normal operating condition.

6.5 Pre and Post Constant Tone

103

In the slowdown phase, turbocharger rotor is decelerated from the maximum rotor speed to the idle condition. During the slowdown, the ring speed ratio does not change too much because the viscosities of both oil films are nearly equal. Therefore, the inner oil whirl frequency decreases from 1,000 to 400 Hz at lowering rotor speeds. Between about 600 and 400 Hz, the post constant tone usually becomes audible in the vehicle cabin, as shown in Fig. 6.9. The low damping effectiveness in the outer oil film is possibly to blame for this audible post constant tone between 600 and 400 Hz. However, it is inaudible at low frequencies (\400 Hz) since the noise level is proportional to frequency squared. The post constant tone in the slowdown phase is considered more critical than the pre constant tone in the run-up phase because engine noise in the slowdown strongly decreases compared to the run-up. This post constant tone could be reduced by active and passive measures that will be discussed in the following section. However, the post constant tone is sometimes confused with the unbalance whistle and other generated noises of components neighboring the turbocharger because their frequencies also decrease with the rotor speed in the slowdown phase. In this case, other noise sources must be at first improved instead of the post constant tone alone. In fact, such unbalance whistle, fuel-pump noise, and high frequency-order engine noises, etc., could be audible in addition to the post constant tone in some car types. Note that the post constant tone level depends on the vehicle type, and how the turbocharger is mounted in the engine.

6.6 Measures of Reducing Constant Tone Level As a result, the inner oil whirl amplitude is proportional to the oil whirling volumetric flow rate given in Eq. (6.18); its whirl frequency xi depends on the volumetric oil flow rate in the axial direction, rotor and bearing ring speeds, as given in Eqs. (6.19) and (6.20). To reduce the constant tone level, the oil whirling flow rate must be kept as small as possible. There are some possible measures to reduce the constant tone level by minimizing the oil whirl amplitude, as shown in Table 6.1.

6.6.1 Active Measures The main reason for the oil whirl is the oil volumetric flow rate surplus in the convergent wedge of the inner bearing clearance. To minimize the constant tone level, the oil flow rate surplus must be relieved to keep the inner oil whirl amplitude as small as possible; therefore, the rotor remains in the stable condition. Equation (6.18) shows that the oil whirling flow rate can be minimized by either reducing the oil flow rate surplus itself (the first term) or increasing the oil flow rate in the axial direction (the second term) to compensate the oil flow rate surplus.

Smaller inner bearing clearance 2c1

Reducing the amplitude of the constant tone ? reduced e = e.c1

Positive effects

Disadvantages

• High oil temperature ? Oil coking • Wear in the mixed lubrication Reducing the amplitude of the constant tone • Small inner bearing surface Shorter inner bearing width Li • Wear in the mixed lubrication Reduced ring speed ratio RSR Reducing the amplitude of the constant tone • Increasing bearing friction at LET • Bad transient behavior at LET Passive Larger outer bearing clearance 2c2 More damping effectiveness in the outer oil film • Larger rotor deflection ? unstable rotor • Bearing modulation ? side bands More damping effectiveness in the outer oil film • Increasing blowby and oil Larger bearing outer diameter Do leakage due to larger seal rings Radial bearing with inner axial grooves Increasing the axial oil flow rate • Bearing wear due to oil insufficiency • Bearing modulation ? side bands Optimizing the turbocharger housing eigenfrequencies Reducing the resonant amplitudes of the turbocharger – housing excited by the constant tone Optimizing stiffness and damping characteristics of Reducing the response amplitudes of components – the car frame and periphery components excited by the constant tone

Active

Table 6.1 Measures to reduce the constant tone level Measures

104 6 Subsynchronous Constant Tone

6.6 Measures of Reducing Constant Tone Level

105

As a result, the oil whirling flow rate decreases, leading to reducing the constant tone level. First, some measures are possible to reduce the oil flow rate surplus according to Eq. (6.18). They are namely, reducing the inner bearing clearance c to increase the bearing stiffness coefficient so that the journal eccentricity e = c.e becomes smaller; shortening the inner bearing width Li; reducing the rotor diameter D (=2R); or decreasing the ring speed ratio. However, all these measures should be thoroughly studied before applying them in turbochargers. They could cause some disadvantages, such as the rotor instability, wear due to the bearing inner surface reduction, increasing bearing friction due to mixed lubrication between the journal and bearing, worsening transient behavior in the low end torque due to decreasing the bearing ring speed, and finally oil coking in the inner bearing clearance. Second, one increases the oil volumetric flow rate in the axial direction by installing a few axial oil grooves in the inner bearing to compensate the oil flow rate surplus. Therefore, the journal whirling amplitude reduces, leading to the constant tone level reduction. However, wear could occur in the bearing because lubricating oil does not sufficiently maintain in the bearing clearance, especially at the start-stop-driving condition or highly unbalanced rotor at the trim balancing.

6.6.2 Passive Measures The oil film damping effectiveness is defined by the ratio of the dissipated energy in the oil film to the total energy of the vibration system per unit of radian. The oil film damping effectiveness results in   1 Ed pcxx2 cx ð6:30Þ Deff  ¼ ¼ 2p Et k 2pð12 kx2 Þ where c is the oil film damping coefficient; k is the rotor-bearing stiffness coefficient;x is the bearing displacement amplitude; x is the bearing whirl frequency. The outer bearing clearance is designed large enough to suppress the constant tone level by increasing the damping effectiveness while transmitting through the outer oil film of the bearing. However, an excessively large outer bearing clearance could induce rotor instability because the bearing stiffness coefficient decreases. Another possible measure is the large bearing outer diameter to increase the outer oil film damping and stiffness coefficients; hence, the damping effectiveness of the outer oil film increases in order to damp and suppress the constant tone level. However, the ring speed ratio of the bearing reduces, leading to the bearing friction increase in the rotor and therefore degrading the turbocharger transient behavior at LET. In addition, the turbocharger housing eigenfrequencies could be optimized to reduce its resonance amplitude excited by the constant tone.

106

6 Subsynchronous Constant Tone

Fig. 6.10 Constant tone levels at min–max inner bearing clearances measured in the bearing housing [4]

Similarly, the stiffness and damping characteristics of the car frame and periphery components neighboring the turbocharger could be improved by shifting their eigenfrequencies to decrease the response amplitudes excited by the constant tone (s. Chapter 7). In the following section, two exemplary measures of reducing the inner bearing clearance (active measure), and increasing the outer bearing clearance (passive measure) are used to reduce the constant tone level of automotive turbochargers. At the active measure, the diametral inner bearing clearance 2c1 is reduced from 15 to 23 lm. As a result, the constant tone level corresponding to the acceleration is decreased from 8 m/s2 at the maximum bearing clearance of 23 lm to nearly 3 m/s2 at the minimum bearing clearance of 15 lm, as shown in Fig. 6.10. This result indicates that the smaller the inner bearing clearance, the lower the constant tone level is. The reason is that the small inner bearing clearance increases the bearing stiffness coefficient; therefore, the journal eccentricity becomes smaller. Hence, the oil whirling volumetric flow rate reduces according to Eq. (6.18), leading to the constant tone level reduction. On the contrary, the journal eccentricity increases at larger inner bearing clearances 2c1 due to lowered bearing stiffness coefficients, leading to higher constant tone levels. Note that, small inner bearing clearances could cause bearing wear and oil coking in the bearing clearance due to hard particle wear and high oil temperatures induced by the bearing friction, respectively. The passive measure of increasing the outer diametral bearing clearance 2c2 leads to decreasing the constant tone level. Measured results in Fig. 6.11 show that

6.6 Measures of Reducing Constant Tone Level

107

Fig. 6.11 Constant tone levels at min–max outer bearing clearances measured in the bearing housing [4]

the larger the outer oil film thickness, the lower the constant tone level is. The large outer oil film thickness induces less bearing friction; the outer oil film temperature is lowered, leading to increasing the damping effectiveness due to large bearing whirl frequency in the outer oil film. Therefore, the outer oil film damps the constant tone level while transmitting through the outer oil film. The bearing friction in the outer oil film increases with reducing the outer bearing clearance 2c2; hence, the oil temperature rises, leading to the oil viscosity reduction in the outer oil film. As a result, the outer oil film damping effectiveness is reduced by decreasing the bearing whirl frequency so that the constant tone level is little damped while passing the outer oil film. As an example, the outer bearing clearance 2c2 is increased from 68 to 84 lm; the constant tone level corresponding to the acceleration response is reduced from about 7 m/s2 at the minimum bearing clearance of 68 lm to nearly 4 m/s2 at the maximum bearing clearance of 84 lm, as displayed in Fig. 6.11. However, the outer oil film stiffness coefficient reduces by increasing outer bearing clearances. Therefore, the rotor amplitude could become unstable in the outer oil film, leading to bearing wear in the mixed boundary lubrication between the bearing rings and bearing center housing. In this case, it should be carefully proved which maximum outer bearing clearance fulfills the rotordynamic stability condition in the operating speed range.

108

6 Subsynchronous Constant Tone

References 1. Hori, Y.: Hydrodynamic lubrication. Springer, (2006) 2. Muszy´nska, A.: Rotordynamics. CRC Press, (2005) 3. Nguyen-Schäfer, H.: Rotordynamics of Automotive Turbochargers. Springer, Berlin-Heidelberg (2012) 4. Nguyen-Schäfer, H., Kleinschmidt, R.: Analysis and nonlinear rotordynamics computation of constant tone in automotive turbochargers, 17th ATK conference. Dresden, Germany (2012)

Chapter 7

Eigenfrequency Modifications to Reduce Constant Tone Level

Constant tone induced by the inner oil whirl is transmitted through the oil films and periphery components neighboring the turbocharger, such as bearing center housing, compressor housing, air filter, charge-air intercooler, exhaust-gas system, and car frame to the vehicle cabin, as shown in Fig. 6.2. While the transmission, the constant tone could excite the periphery components, especially nearby the turbocharger. The resonance amplitudes occur when the component eigenfrequencies xc;i and xc;k equal the excitation frequencies of the inner oil whirl xIOW;min and xIOW;max , as shown in Fig. 7.1. The response amplitudes of the periphery components additionally amplify the airborne noise level at the bandwidth Df of the constant tone frequency. The audible noises are usually considered as the constant tone, especially in the run-up and slowdown phases at the frequency range between 400 and 600 Hz, as discussed in Sect. 6.5. Thus, the noise levels strongly depend on the vehicle characteristics; as an example, the constant tone is inaudible in one car type; however, it could be audible in another car type even with the same turbocharger. In this case, the eigenfrequencies of the vehicle types are to blame for the airborne acoustic quality. The component eigenfrequency is calculated from the component stiffness coefficient K and mass m [1, 4]. rffiffiffiffi K xc ¼ ð7:1Þ m Figure 7.1 shows the Campbell diagram of a periphery component nearby the turbocharger. Every component eigenfrequency in the synchronous excitation line 1X equals the constant tone frequency; therefore, many resonances take place at the cutting points between the component eigenfrequencies and constant tone frequency bandwidth. The periphery component resonance amplitudes intensify the airborne in the constant tone frequency bandwidth, as seen in Fig. 7.1. In this case, the resonance amplitudes of the excited components are also audible in the vehicle cabin, but not alone the constant tone induces the airborne noise.

H. Nguyen-Schäfer, Aero and Vibroacoustics of Automotive Turbochargers, DOI: 10.1007/978-3-642-35070-2_7, Ó Springer-Verlag Berlin Heidelberg 2013

109

110

7 Eigenfrequency Modifications to Reduce Constant Tone Level

Fig. 7.1 Campbell diagram of periphery components

Excitation frequency bandwidth Δf

ω

resonance

resonance

ωc ,max

1X

ω c,k

component eigenfrequency bandwidth

ωc,i

ωc ,min ω IOW ,min ω IOW ,max

ωexc

To reduce or avoid the intensified airborne noise, the component eigenfrequencies should be moved out of the constant tone excitation frequency range so that no resonance takes place in the constant tone frequency bandwidth. According to Eq. (7.1), one can vary the component eigenfrequency by either stiffening the stiffness coefficient or reducing the component mass. As an example, the stiffness coefficient is increased so that the minimum component eigenfrequency exceeds the maximum constant tone frequency, as shown in Fig. 7.2. Thus, the response amplitude is far from the resonance frequency and remains small, leading to lowered airborne noise level. To increase the eigenfrequencies, the periphery components should be tightened and damped in the engine, or their wall thickness is increased. In the case of a long flexible air pipe, it should be fixed at some damping supports between the pipe ends to increase its stiffness coefficient. However, it is not always easy to stiffen all periphery components in the engine. Fig. 7.2 Campbell diagram of optimized periphery components

ω

Excitation frequency bandwidth Δf

1X

ω c ,max

ω c ,min small amplitude

ω IOW ,min

ω IOW ,max

ωexc

7 Eigenfrequency Modifications to Reduce Constant Tone Level

111

12

ζ≡ 10

c 2 Km

ζ = 0.05 8

|A|/ ε

Amax

ζ = 0.10

6

4

ζ = 0.25 ζ = 0.50 ζ = 1.00

2

0

0

0.5

1

1.5

2

2.5

3

η = Ω/Ωcr Fig. 7.3 Response amplitudes at various damping ratios f

To detect noise of the engine components, a 3-D acoustic camera is used to analyze and visualize the airborne noise field of the engine in different operating conditions. Noise pressure levels are recorded by means of microphone arrays, such as ring-shaped, spherical, or near-field planar arrays with to about 120 microphones. By using the analyzing and evaluating softwares, e.g., Software Noise Image, the noise field-induced by the turbocharger itself and its periphery components are visualized; the noise spectra diagram (Waterfall plot) is additionally displayed. This method is useful to detect which component stiffness should be improved in the applied vehicle type. Alternatively, the periphery components could be damped in order to reduce the response amplitude j Aj even at the resonance g ¼ 1, as shown in Fig. 7.3. Note that the more the damping ratio is, the lower the vibration amplitude responses. The dimensionless damping ratio f is defined by the ratio between the component stiffness, damping coefficients, and mass [2, 3]. c f ¼ pffiffiffiffiffiffiffi 2 Km

ð7:2Þ

In practice, the excited components are damped by elastomeric materials at the supports in order to reduce the response amplitude. In fact, acoustic engineers have to take the compromise between the periphery component improvement and costs; rotor dynamic engineers have to reduce the induced noise in automotive turbochargers.

112

7 Eigenfrequency Modifications to Reduce Constant Tone Level

References 1. Muszy´nska, A.: Rotordynamics. CRC Press, New York (2005) 2. Nguyen-Schäfer, H.: Rotordynamics of automotive turbochargers. Springer, Berlin-Heidelberg (2012) 3. Vance, J.: Rotordynamics of turbomachinery. Wiley, New York (1988) 4. Vance, J., Zeidan, F., Murphy, B.: Machinery vibration and rotordynamics. Wiley, New York (2010)

Conclusions

Besides the functionality of automotive turbochargers, the airborne noise level of turbochargers has recently become a much more important topic, especially in passenger vehicles. To analyze the noise sources of turbochargers, two flow instabilities in the compressor have been studied in this book. First, the airflow instability in the compressor causes rotating stall and surge, which induce the growling and whining noise, respectively. Second, the self-excited instability of the inner oil whirl in the rotating floating ring bearings, leading to the constant tone (howling). In addition to these noise types, the residual unbalance after trim balancing and unbalance changes after a long-term operating period generates the unbalance whistle. All these are the noise sources induced in automotive turbochargers of passenger vehicles. These uncomfortable noise types are transmitted from the noise sources through the turbocharger itself, periphery components nearby the turbocharger, and vehicle frame to the passenger cabin. The noise pressure levels strongly depend on the car types; they are subjectively evaluated by different drivers and passengers for the different car types in various operating conditions. The noise computational simulation of automotive turbochargers is much more complex and difficult than computational fluid dynamics (CFD) because the noise computational simulation involves many interdisciplinary simulations, such as the noise generation simulation using CFD and computational nonlinear rotordynamics (CNR) in the near-field, acoustic propagation simulation in the periphery components neighboring turbochargers using computational aeroacoustics (CAA) including linearized Euler’s equations (LEE) for linear aeroacoustics, and acoustic perturbation equations (APE) using the Lighthill’s equation for nonlinear aeroacoustics in the mid-field, and Far-Field simulation of noise radiation into the environment outside the vehicle using the Ffowcs Williams and Hawkings equations (FWHE), which is based on the integral methods using finite element method (FEM) and boundary element method (BEM) for solving the noise pressures of the nonlinear FWHE and Kirchhoff-Helmholtz integral equations. Mostly, the BEM is used to solve the Kirchhoff-Helmholtz integral

H. Nguyen-Schäfer, Aero and Vibroacoustics of Automotive Turbochargers, DOI: 10.1007/978-3-642-35070-2, Ó Springer-Verlag Berlin Heidelberg 2013

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114

Conclusions

equation with aero- and rotordynamic noise sources by summation of the induced noise pressures over the boundary surface for three-dimensional acoustic field problems. Therefore, it is a big challenge in the near future to develop a commercial computational code for nonlinear aeroacoustics, fluid dynamics, and nonlinear rotordynamic simulations of automotive turbochargers in which the periphery components nearby turbochargers are taken into account in passenger vehicles. As a matter of fact, the acoustic evaluation usually comes at the end of the product development phase because the new vehicle model has never existed long before its launching on the market. Therefore, airborne noise could not be evaluated in the hot gas test rig in advance since the acoustic quality in the vehicle cabin depends on the different influences of the vehicle frame and periphery components nearby turbochargers. Sometimes, modified radial bearing types must be applied in a certain new car model although they have not yet been thoroughly evaluated for durability and wear in a long-term operating period. For this reason, the airborne noise should be deeply understood so that one can react and improve much more quickly at the end of the development phase, just shortly before the start of production (SOP) for every customer’s vehicle model.

Further Reading

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11.

12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

Bartel, D.: Simulation von Tribosystemen. Springer-Teubner, German (2010) Cremer, L., Möser, M.: Technische Akustik (in German), 5. Auflage, Springer, Berlin (2003) Crocker, M.J.: Noise and vibration control. Wiley, Inc. (2007) Cumpsty, N.A.: Compressor aerodynamics. Krieger Publishing Company, New York (2004) Fahy, F., Gardonio, P.: Sound and structural vibration, 2nd edn. Academic, New York (2007) Greitzer, E.M.: The stability of pumping systems, the 1980 Freeman scholar lecture. J. Fluids Eng. 103 (1981) Hori, Y.: Hydrodynamic lubrication. Springer, Berlin (2006) Muszy´nska, A.: Rotordynamics. CRC Press, Boca Raton (2005) Nguyen-Schäfer, H.: Rotordynamics of automotive turbochargers. Springer, Heidelberg (2012) Nguyen-Schäfer, H., Kleinschmidt, R.: Analysis and nonlinear rotordynamics computation of constant tone in automotive turbochargers. In: Proceedings of the 17th ATK Conference, Dresden, Germany (2012) Nguyen-Schäfer, H., Sprafke, P.: Numerical study on interaction effects of the bubbles induced by air-release and cavitation in hydraulic systems, tenth bath international fluid power workshop, Bath, UK. Research Studies Press Ltd., Hertfordshire, England (1997) Nguyen-Schäfer, H.: Nonlinear rotordynamics computations of automotive turbochargers using rotating floating ring bearings at high rotor speeds. In: Proceedings of the 10th SIRM International Conference, Berlin, Germany (2013) Nordmann, R.: Schwingungsberechnung von nichtkonservativen rotoren mit hilfe von links—und rechts-eigenvektoren (in German), vol. 269. VDI Bericht (1976) Rieger, N.F.: Rotordynamics 2—problems in turbomachinery, CISM courses and lectures No. 297 (1988) Rieger, N.F.: Balancing of rigid and flexible rotors. U.S. DoD (1986) Schmied, J.: Program MADYN 2000 for computation of rotordynamics, Delta J.S., Zurich (2011) Schobeiri, M.: Turbomachinery flow physics and dynamic performance, 2nd edn, Springer, Berlin (2012) Vance, J.: Rotordynamics of turbomachinery. Wiley Inc., New York (1988) Vance, J., Zeidan, F., Murphy, B.: Machinery Vibration and Rotordynamics. Wiley, New York (2010) Whitfield, A., Baines, N.C.: Design of radial turbomachines, Pearson education, Longman scientific and technical (1990) Aris, R.: Vectors, tensors, and the basic equations of fluid mechanics. Dover Publications Inc., New York (1989) Synge, J.L., Child, A.: Tensor calculus. Dover Publications Inc., New York (1969)

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Appendix A Thermodynamic Characteristics of Perfect Gases

Some essential thermodynamic characteristics of gases are required to be known in turbocharging. They have usually been applied to the turbocharging of downsized engines where the charge air and exhaust gas are assumed as perfect gases. Total temperature Tt in Kelvin (K) is resulted from the sum of the static Ts (K) and dynamic temperatures Tdyn (K). The static temperature is measured at the wall where the gas velocity equals zero due to the viscous boundary layer. Tt ¼ Ts þ Tdyn ¼ Ts þ

v2 2cp

where v is the gas velocity; cp is the heat capacity at constant pressure. Total pressure pt is calculated from the isentropic gas equation of  j j 1 pt ¼ ps TTst  j 2 j1 ¼ ps 1 þ j1 2 M where ps is the static pressure; j = cp/cv is the isentropic exponent of gas; cv is the heat capacity at constant volume; M = v/c is the Mach number of gas; c = the sound speed is calculated with the gas constant Ra. pffiffiffiffiffiffiffiffiffiffiffiffi c ¼ jRa Ts

ðA:1Þ

ðA:2Þ

ðA:3Þ

Specific total enthalpy ht is resulted from the sum of the gas specific enthalpy and specific kinetic energy of gas.

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118

Appendix A: Thermodynamic Characteristics of Perfect Gases

ht ðTÞ ¼ hðTÞ þ

v2 2

ðA:4Þ

The gas specific enthalpy h (enthalpy per mass unit, J/kg) is defined by hðTÞ ¼ cp ðT  T0 Þ p ¼ uðTÞ þ q ¼ uðTÞ þ Ra T

ðA:5Þ

within q is the gas density; T0 is the reference temperature; u is the gas specific internal energy. The gas specific internal energy equals the product of the heat capacity at constant volume cv and temperature difference of gas DT. uðTÞ ¼ cv ðT  T0 Þ  Ra T0 ¼ cv DT  Ra T0

ðA:6Þ

where the gas enthalpy at T0 = 273.14 K (0 °C) is defined by hðT0 Þ  0

ðA:7Þ

Thus, one obtains the gas-specific internal energy according to Eq. (A.7) uðT0 Þ  Ra T0

ðA:8Þ

Appendix B Transformation of Coordinates

The discretized noise pressure equations in the local orthogonal coordinate system (n1, n2, n3) of the finite element grid must be transformed into the global Cartesian coordinate system (x1, x2, x3) in the FEM computation, as shown in Fig. B.1. To carry out this transformation, some necessary mathematical backgrounds are briefly described. More details can be found in many literatures [17, 21, 22]. The position vector r in the control volume V is written in • the Cartesian coordinate system (x1, x2, x3) r ¼ x1 e1 þ x2 e2 þ x3 e3 :

ðB:1Þ

• the curvilinear coordinate system (n1, n2, n3) r ¼ n1 g1 þ n2 g2 þ n3 g3 :

Fig. B.1 Transformation of coordinates

ðB:2Þ

x3

Cartesian coordinate system (x1, x2, x3)

ξ3

V

Curvilinear coordinate system (ξ1, ξ2, ξ ) 3

dV g3 g3

g1 ξ1

g1

e3 e1

e2

r

x2

g g2 2

x1

H. Nguyen-Schäfer, Aero and Vibroacoustics of Automotive Turbochargers, DOI: 10.1007/978-3-642-35070-2, Ó Springer-Verlag Berlin Heidelberg 2013

ξ2

119

120

Appendix B: Transformation of Coordinates

where e1, e2, and e3 are the covariant basis vectors (unit vectors) of the Cartesian coordinate system; g1, g2, and g3 are the covariant basis vectors (non-unit vectors) of the curvilinear coordinate system; xi is the covariant vector component of r for i = 1, 2, 3; ni is the covariant vector component of r for i = 1, 2, 3. The covariant basis vectors tangent to the curvilinear coordinate are defined by using Einstein summation symbol. or oðxj ej Þ ¼ oni oni 3 X oxj oxj  ej ¼ ej oni oni j¼1

gi ¼

ðB:3Þ

The contravariant basis vectors normal to the surfaces (nj, nk) of the curvilinear coordinate system are given by gj 

3 X onj onj em ¼ em oxm ox m m¼1

ðB:4Þ

The scalar product of the co- and contravariant basis vectors is given by gk g j 

onj oxi onj :  djk ei em ¼ onk oxm onk

ðB:5Þ

where dkj is the Kronecker delta; dkj = 0 if j = k; dkj = 1 if j = k. The metric coefficients of the curvilinear coordinate system are defined by • Covariant metric coefficient:

gij ¼ gi :gj

ðB:6Þ

gij ¼ gi :g j

ðB:7Þ

gij ¼ gi :g j

ðB:8Þ

• Contravariant metric coefficient:

• Mixed metric coefficient:

A fluid vector v can be formulated in the curvilinear coordinate system

Appendix B: Transformation of Coordinates

121

v ¼ v1 g1 þ v2 g2 þ v3 g3  vi gi ðcovariant basis vectorsÞ

¼ v1 g1 þ v2 g2 þ v3 g3  vi gi ðcontravariant basis vectorsÞ

ðB:9Þ

where v1, v2, v3 are the contravariant vector components (raised indexes); v1, v2, v3 are the covariant vector components (lowered indexes). The Jacobian determinant is applied to transform the coordinates and are given by  ox1 ox1 ox1        on1 on2 on3  oxi oxj oxk  ox2 ox2 ox2  J   on1 on2 on3  ¼ eijk ; 8 i; j; k ¼ 1; 2; 3: ðB:10Þ  ox ox ox  on1 on2 on3 3 3   3 on1

on2

on3

within eijk is the permutation symbol defined by 8 if i ¼ j or j ¼ k; or k ¼ i < 0 eijk  1 if ijk is an even permutationð¼ 123Þ : 1 if ijk is an odd permutationð¼ 132Þ

ðB:11Þ

In an orthogonal basis, the Jacobian determinant is resulted from Eqs. (B.3), (B.6), and (B.10). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J ¼ g11  g22  g33 ðB:12Þ

Using Eq. (B.10), the infinitesimal volume dV is computed in the curvilinear coordinate system. dV  dx1 dx2 dx3

¼ dX 1 ðdX2  dX3 Þ    oxi oxj oxk dn : dn : dn ¼ eijk on1 1 on2 2 on3 3

ðB:13Þ

¼ J:ðdn1 dn2 dn3 Þ

where the vector dXi is defined by h oxi oxi dXi ¼ on on2 1

oxi on3

iT

; 8i ¼ 1; 2; 3:

ðB:14Þ

The velocity v does not change at changing the coordinate system. Therefore, the volume integral of the velocity v depending on xi = xi (n1, n2, n3) for i = 1, 2, 3 over the control volume V is resulted from Eq. (B.13). Z Z vðx1 ; x2 ; x3 ÞdV ¼ vðn1 ; n2 ; n3 ÞJ:dn1 dn2 dn3 ðB:15Þ V

V

Similarly, the surface integral of a function u(X) is calculated over the tangential surface ST in the curvilinear coordinate system, where X(k1, k2)

122

Appendix B: Transformation of Coordinates

parameterizes the surface S in the Cartesian coordinate system.  Z Z oX oX  uðXÞn:dS ¼ u½Xðk1 ; k2 Þ: dk1 dk2 ok1 ok2

ðB:16Þ

ST

S

The coordinate transformation gives the operators written in Einstein summation convention in the curvilinear coordinate system: • Gradient operator of a scalar function of u = u(n1, n2, n3) ru ¼

o/ i g  u;i gi oni

ðB:17Þ

• Divergence operator of a vector v = v(v1, v2, v3) o ðg v j Þgi ¼ r:v ¼ oni j ¼

v;ij

k

þv

Cikj



 j  ogj j ov v þ g gi oni oni j

ðB:18Þ

where the first term on the rhs of Eq. (B.18) defines the derivative of vj with respect to ni. v;ij 

ov j oni

ðB:19Þ

The Christoffel symbol of the second kind is defined by Cikj 

o2 xm onj : oni onk oxm

ðB:20Þ

• Curl operator of a vector v = v(v1, v2, v3) rv

¼ gk ono k  ðgi vi Þ ¼ gk  gi vi;k þ gk  gi;k vk

¼ ekim gm vi;k

ðB:21Þ

where ekim is the permutation symbol given in Eq. (B.11); The third term on the rhs of Eq. (B.21) defines the derivative of vi with respect to nk. vi; k 

ovi onk

ðB:22Þ

Appendix C Solutions of the Characteristic Equation with Complex Coefficients

The characteristic quadratic equation with complex coefficients is derived from the vibration equation. DðsÞ  s2 þ ða þ jbÞs þ ðc þ jdÞ ¼ 0

ðC:1Þ

where s ¼ ða  jxn Þ 2 C are the complex eigenvalues;

a; b; c; d 2 R are the real numbers:

By solving the characteristic equation D(s) = 0, the eigenvalue s results in ffi  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  a þ jb a þ jb 2 s1; 2 ¼  ðc þ jdÞ ¼ a  jxn ðC:2Þ  2 2 After calculating and rearranging the real and imaginary terms of the eigenvalue, one obtains [8] 2 31 2 2  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2  a 1 4 a2  b2 a  b2 ab d 5 a ¼   pffiffiffi c þ c þ ðC:3Þ 2 2 4 4 2

and

2 31 2 2  2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2  b 1 4 a  b2 a  b2 ab d 5 c þ c þ xn ¼  pffiffiffi  2 2 4 4 2

ðC:4Þ

Necessary condition for the rotordynamic stability is that the real term a of the eigenvalue must be negative.

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124

within

Appendix C: Solutions of the Characteristic Equation with Complex Coefficients

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii12 a 1 h a ¼   pffiffiffi E þ E2 þ F 2 \ 0 2 2 E

 2  2 a  b2 a  b2 c ¼ þc 4 4  ab d F 2

ðC:5Þ

ðC:6Þ ðC:7Þ

Thus, the stability condition (a \ 0) for the rotor given in Eq. (C.5) becomes after a few calculation steps. a ðac þ bdÞ  d2  0

ðC:8Þ

Appendix D Stability Condition of Linear Aeroacoustics

The eigenvalues analysis is based on the free vibration response of the linear vibration. The free vibration response is resulted by superimposing the harmonic components of the linear aeroacoustic system without excitation of noise source. In fact, the free noise vibration response is exactly the homogeneous solutions of the vibration equations. Each harmonic component has a complex eigenvalue in which the imaginary part is the eigenfrequency (natural frequency) of the vibration system; the real part indicates the stability characteristic of the vibration system. The free noise vibration response of the system is written in rðtÞ ¼

N X

ri e k i t

i¼1

ðD:1Þ

where ri is the response amplitude corresponding to the eigenvalue; ki is the complex eigenvalue of the eigenmode i; N is the number of degrees of freedom (DOF) of the vibration system. The complex eigenvalue consists of the real and imaginary parts. k ¼ a  jxd

ðD:2Þ

within a is the real part of the eigenvalue, called the growth/decay rate; xd is the imaginary part of the complex eigenvalue (eigenfrequency or damped natural frequency). Substituting k into Eq. (D.1), one obtains the vibration free response by applying Euler’s theorem.

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126

Appendix D: Stability Condition of Linear Aeroacoustics

rðtÞ ¼ ¼

N X i¼1

N X i¼1

ri;fw eai t ejxi t þ

N X

ri;bw eai t ejxi t

i¼1

ri;fw eai t ½cosðxi tÞ þ j sinðxi tÞ þ

 xðtÞ þ jyðtÞ

N X i¼1

ri;bw eai t ½cosðxi tÞ þ j sinðxi tÞ ðD:3Þ

The parts in the square brackets [ ] in Eq. (D.3) are the harmonic vibrations with the eigenfrequencies of the forward and backward frequency components. Hence, the maximal amplitude is limited at 1 at any time t. On the contrary, the terms in front of the square brackets [ ] containing an exponential function exp (ait) change with time. The amplitude of the noise response depends on the growth/decay rate ai. If the rate ai is positive, the noise amplitude increases with time without limit. The noise response is unstable at any positive rate a; therefore, a is the key parameter for the stability analysis of the noise in the linear aeroacoustics. Cases of the growth/decay rate a for the noise response behavior: • a \ 0: stability; • a = 0: onset of instability (threshold of instability); • a [ 0: instability. At a decay rate (a \ 0), the vibration amplitude with the frequency x diminishes in a short time. The envelope of the vibration is the exponential function r0.exp (at) that begins from the initial position r0 at time t = 0 and decreases with time to zero. The corresponding forward orbit decays from the initial position r0 to zero as t??. On the contrary, at a growth rate (a [ 0), the vibration begins at the initial position and increases with time exponentially; therefore, the vibration is unstable. At the rate of a = 0, the vibration amplitude is unchanged with time, neither decreasing nor increasing with time because the amplitude of the function r0.exp (at) always remains constant at r0 at any time t. The noise vibration moves in a circular orbit that is called limit cycle, which occurs in the Hopf bifurcation. The vibration is at the threshold of instability (onset of instability). To study noise vibration stability, the complex eigenvalues k are analyzed in the k plane, as shown in Fig. D.1. The complex eigenvalue and its conjugated eigenvalue have the common real part and opposite imaginary part with positive and negative signs that correspond to forward and backward damped natural frequencies. If both these eigenvalues lie in the half-left plane (a \ 0), the noise vibration behavior is stable; on the half-right plane (a [ 0), it is unstable. In case of the threshold of instability (a = 0), the noise vibration moves with the damped natural frequency xd at the constant amplitude, neither decreasing nor increasing with time.

Appendix D: Stability Condition of Linear Aeroacoustics

127

Fig. D.1 Stability analysis in the k plane

In order to maintain the noise vibration in a stable condition, all eigenvalues of the vibration system must lie on the left half-plane, so that the growth/decay rates are always negative. In this case, there are two possibilities, as shown in Fig. D.1: • When they locate in the upper left half-plane, the vibration response is a stable forward whirl and decays with time from the initial position to zero. • When they locate in the lower left half-plane, the vibration response is a stable backward whirl and decays with time from the initial position to zero, but the whirl direction is opposite to the forward whirl.

Appendix E Routh-Hurwitz Stability Criterion

Characteristic equation is a polynomial equation of nth order. DðkÞ ¼ an k n þ an1 k n1 þ . . . þ a2 k 2 þ a1 k þ a0 ¼ 0

ðE:1Þ

where k are the complex eigenvalues with k = a + jxd; ai is the equation coefficients, i = 0, 1, 2,…, n; n is the polynomial exponent that equals twice DOF (n = 2 N). Having used the Gauss theorem, Eq. (E.1) can be written in the polynomial with the n order of k where ki are the roots of D(k). DðkÞ ¼ an ðk  k1 Þðk  k2 Þ    ðk  kn1 Þðk  kn Þ

ðE:2Þ

Finding the roots ki of the polynomial D(k) in Eq. (E.1) is quite difficult, especially at a large number of DOFs. In fact, it need not be known the exact values of the roots ki, but only the real parts a of the eigenvalues are necessary for the stability analysis. In this case, the Routh-Hurwitz stability criterion is applied to analyze the eigenvalues in linear vibration system. According to the Routh-Hurwitz criterion, the vibration response is stable when the following two conditions are satisfied: 1. All coefficients ai in the characteristic equation (E.1) must be nonzero and have the same positive or negative signs (necessary condition). 2. Hurwitz determinant Dn-1, and its diagonal determinants Di for i = 1, 2,…, n-2 must be positive (sufficient condition).

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130

Appendix E: Routh-Hurwitz Stability Criterion

Dn1

  an1   an   0  ¼  0  0   0   0

an3 an2 an1 an 0 0 0

an5 an4 an3 an2 : : :

: : an5 an4 : : :

... : ... : ... : ... : . . . a1 . . . a2 a5 a3

 0  0  :  :  [ 0 0  a0  a1 

ðE:3Þ

where the coefficients an-i equal zero at all negative indices (n-i) for i = 0, 1,…, n. The Hurwitz determinant Dn-1 and diagonal determinants Di for the characteristic equation D(k) of n order (n = 2 DOF) can be written in D1 ¼ jan1 j [ 0;    an1 an3   [ 0;  D2 ¼  an an2     an1 an3 an5    D3 ¼  an an2 an4  [ 0. . .  0 an1 an3 

Dn2

  an1   an   0 ¼  0   0  0

an3 an2 an1 an 0 0

an5 an4 an3 an2 : :

... ... ... ... ... ...

: : : : a3 a4

 0  0  0  [ 0: a0   a1   a2

About the Author

Dr. Hung Nguyen-Schäfer is a senior expert in development of turbochargers at Bosch Mahle Turbo Systems (BMTS) in Germany. He received his B.Sc. and M.Sc. in Mechanical Engineering focusing on nonlinear vibrations in fluid mechanics from the University of Karlsruhe, Germany in 1985 and his Ph.D. in nonlinear thermoand fluid dynamics from the same university. In 1988 he joined the Bosch company and worked as a technical manager on many development projects. Since 2007, he has been in charge of rotordynamics, bearing, and designing platforms of automotive turbochargers at Bosch Mahle Turbo Systems (BMTS) located in Stuttgart, Germany, the joint venture of Bosch and Mahle. He is also the author of Rotordynamics of Automotive Turbochargers, Springer, 2012.

H. Nguyen-Schäfer, Aero and Vibroacoustics of Automotive Turbochargers, DOI: 10.1007/978-3-642-35070-2, Ó Springer-Verlag Berlin Heidelberg 2013

131

Index

A Absolute velocity, 51 Acceleration amplitude, 67 Acoustic camera, 111 Acoustic perturbation equations (APE), 27, 113 Acoustic propagation, 22 Aerodynamic noise, 3 Airborne noise, 3, 97 Airflow reversal, 53 Angular momentum, 99 Angular momentum equation, 49 Angular velocity, 99 Average fluid velocity ratio, 92

B Backward swept, 47 Backward whirl, 127 Ball passing frequency, 71 BEA, 32 Bearing filling grade, 76 Beat phenomena, 41 Biturbocharger, 6 Blade outlet, 47 Boundary element method, 27 Bubble fraction, 75 Bunsen’s coefficient, 76

C CAA, 28 Cameron and Vogel’s equation, 77 Campbell diagram, 109 Carrier frequency, 38, 39, 42 CFD, 27, 113

Characteristic equation, 60, 123, 129, 130 CHIEF, 32 CHRA, 2 Christoffel symbol, 122 Circumferential velocity, 51 Complex dynamic stiffness, 66 Complex eigenvalue, 123, 125, 126 Complex transfer impedance, 66 Compression wave, 16 Compressor, 2 Compressor characteristic slope, 57 Compressor specific work, 50 Compressor surge, 53 Compressor tongue, 46 Compressor volute, 49 Compressor wheel, 45 Computational nonlinear rotordynamics (CNR), 27, 113 Computational rotordynamics, 27, 113 Conical mode, 81 Constant tone, 3, 12 Contravariant basis vectors, 120 Corrected mass flow rate, 54 Couple unbalance, 64, 65 Covariant basis vectors, 120 Crackling noise, 12 Cross-coupled damping coefficient, 93 Cross-coupled stiffness coefficient, 91 Cylindrical mode, 81

D Damping ratio, 111 Decay rate, 125, 126 Deep surge, 53 De Haller’s rule, 51

H. Nguyen-Schäfer, Aero and Vibroacoustics of Automotive Turbochargers, DOI: 10.1007/978-3-642-35070-2, Ó Springer-Verlag Berlin Heidelberg 2013

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Index

134

D (cont.) Diffuser, 53 Dirichlet boundary condition, 30 Discrete Fourier Transform (DFT), 81 DNS, 28 Downsized engine, 2 Downsizing, 1 DPF, 15 Dynamic instability, 58 Dynamic temperatures, 117 Dynamic unbalance, 65 Dynamic viscosity, 98, 100, 101

E Eccentricity, 64, 65 Eigenfrequency, 109, 125 Emission law, 1 Engine noise, 103 Euler’s turbocharger equation, 51 Exhaust gas recirculation, 4 Exhaust-gas turbochargers, 2 Expansion wave, 16

F Far-Field simulation, 113 FEM, 32 Ffowcs Williams and Hawkings’ equation (FWHE), 27, 113 First turbocharger equation, 50 Floating ring bearings, 97, 101 Flow instabilities, 54 Flow separation, 51 Forward propagating pressure, 20 Forward whirl, 127 Free vibration response, 125 Fundamental train frequency, 70

G Galerkin formulation, 32 Gauss divergence theorem, 33 Green’s second theorem, 29 Growling noise, 11

H Heat capacity, 117 Helmholtz equation, 29 Helmholtz frequency, 61

Henry-Dalton’s law, 76 Highland speed reserve, 54 High-order harmonic noise, 12 High-pressure EGR, 4, 5 Homogeneous solutions, 125 Hopf bifurcation, 126 Howling, 12, 68 HPDI, 1 Hub, 48

I Impeller, 48 Inlet flow angle, 51 Instability, 126 Inverse square law, 27 Isentropic coefficient, 18 Isentropic exponent, 117

K Kirchhoff-Helmholtz integral equation, 27, 30, 113 Kronecker delta tensor, 25

L Laplacian operator, 24 Lighthill’s analogy, 25 Lighthill’s equation, 25 Lilley and Dowling’s equation, 25 Limit acceleration responses, 66 Limit cycle, 68, 126 Linear rotordynamics, 136 Linearized Euler’s equations (LEE), 27, 113 Low-cycle fatigue, 5 Low-end torque, 6, 54, 98 Low-pressure EGR, 5 Low-speed balancing, 63 LP-EGR, 1 LSB, 39, 40, 42, 43

M Mach number, 117 Meridional velocity component, 51 Mild surge, 56 Misalignment, 41, 65 Misalignment angle, 65 Modulations, 37, 39–41, 43, 70 mpg, 3

Index N Nabla operator, 23 Neumann boundary condition, 30 Noise impedance, 20 Noise intensity, 20 Noise power, 21 Noise power level, 21 Noise pressure level, 22 Noise source strength, 27 Nonlinear aeroacoustics, 35 Nonlinear rotordynamics, 68

O Oil mixture density, 76 Oil whirl, 39, 41 Onset of instability, 126 Outlet blade angle, 51 Overall turbocharger efficiency, 50

P Perturbed density, 17 Perturbed medium density, 18 Perturbed medium temperature, 18 Perturbed noise pressure, 17, 18 Perturbed temperature, 17 Pitch diameter, 70 Pressure ratio , 4, 49, 54 Pressure side, 52 Propagating noise pressure, 23 Propagating wave front, 15 Pulsation whistle, 11

R Radial bearings, 2 Radial compressor wheel, 47 Rayleigh-Ritz method, 33 Residual unbalance, 65 Resonance, 68, 69, 97, 111 Reynolds lubrication equation, 77, 99 Reynolds number, 35 Rigid rotors, 63 Ring speed ratio, 93, 99–101 Robin boundary condition, 30 Rolling-element bearings, 70 Rotating floating ring bearings, 101 Rotating stall, 54 Rotational noise, 11 Rotordynamic noise, 3 Routh-Hurwitz criterion, 61, 129

135 S Shop balancing, 63, 65 Shroud , 48 Single-stage turbocharger, 4, 5, 7, 8 Sommerfeld number, 80 Sonic speed, 117 SOP, 114 Sound speed , 17 Specific enthalpy, 117, 118 Specific internal energy, 118 Specific kinetic energy , 117 Stability , 123, 125, 126, 136, 137 Static instability, 58 Static temperature, 117 Static unbalance, 64, 65 Statically stable, 58 Statically unstable, 58 Sturm-Liouville problems, 30 Subsynchronous, 72 Subsynchronous noise, 89 Suction side, 52 Super back, 48 Supersynchronous, 35, 72 Synchronous and asynchronous noise, 90

T Threshold of instability, 56, 126 Threshold of static instability, 58 Throttle characteristic , 56 Throttle characteristic slope, 57 Thrust bearing, 2 Total enthalpy, 117 Total pressure, 117 Total temperature, 117 Transfer impedance , 36, 66 Trim balancing, 64, 66–68 Turbo-compound, 9 Two-stage turbochargers, 4, 7, 8 Two-times ball spin frequency, 71

U Unbalance change, 66 Unbalance moment, 65 Unbalance whistle , 11, 63, 64, 66, 67 USB, 38, 40, 41, 43

V VCR, 1 Velocity triangle map, 51 Vibration modes, 81 Volume acceleration, 27

136

V (cont.) Volume velocity, 26 VVT, 1

W Waterfall plot, 35, 41, 81 Wave equation, 24

Index Wavelength, 26 Wavenumber, 26 Wear noise, 12 Whining noise, 11